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WORKS OF 

PROF. C. L. CRANDALL, C.E., 

PUBLISHED BY 

JOHN WILEY & SONS. 


THE TRANSITION CURVE. 

By Offsets and by Deflection Angles. Second Edi¬ 
tion, Revised and Enlarged. i6mo, vi+99 pages, 
14 figures. Morocco, $1.50. 

RAILWAY AND OTHER EARTHWORK TABLES. 

8vo. Cloth, $150. 

TEXT B 3 CK ON GEODESY AND LEAST 
SQUARES. 

Prepared for the Use of Civil Engineering 
Students. 8vo, x + 329 pages, 102 figures. Cloth, 
$3 00. 




THE TRANSITION CURVE 


\ 

BY OFFSETS AND BY DEFLECTION 
ANCLES. 


BY 

C. L. CRANDALL, C.E., 

Professor of Railway Engineering, Cornell University; 
Member American Society of Civil Engineers. 


SECOND EDITION, REVISED AND ENLARGED. 

FOURTH THOUSAND. 


NEW YORK: 

JOHN WILEY & SONS. 
London: CHAPMAN & IIALL, Limited. 
1909 . 



Copyright, 1899, 

BY 

C. L. Crandall. 






f i 

M I - 

4 


U 


01jr ©rirntifir flrrHfl 
Robert Srummnttb attb doatpattg 
Hrai Herb 



PREFACE. 


Much lias been written upon the subject of .transition curves 
within the past few years, and the theory and practice have 
been accurately worked up for small central angles, notably 
in the “ Holbrook Spirals.” Accurate methods are here, it is 
believed for the first time, developed for the true transition 
curve, curvature increasing directly with the distance, which 
will hold for large central angles for both the offset and the 
deflection methods. The extension of the limits to include 
long transition curves with large central angles increases the 
flexibility of the alignment of a railroad, which will tend to 
economy of construction in all but the most easy country. 

The offset method is worked up for use during location and 
the greater part of construction on account of its extreme sim¬ 
plicity ; the increased labor over that required for simple 
curves being almost insignificant. 

The method by deflections with transit and chain will usually 
be preferred for curves over 200 to 300 feet in length, in run¬ 
ning centers on the finished roadbed, for laying track, or 
other accurate work. Table III will give the deflections for 
any true transition curve connecting a circular curve with a 
tangent, or connecting the two branches of a compound curve, 
with accuracy for all central angles of transition curve up to 
60°. The deflections taken out for the most common curves 
can be conveniently tabulated on the blank pages inserted for 
the purpose. 

While intended primarily for the use of civil engineering 
students, it is believed that the complete set of tables given 
will render the methods at least as rapid and convenient in 
actual use in the field as the more restricted or approximate 
ones now in use. 

Special acknowledgment is due to H. N. Ogden, C.E., 
formerly an instructor here, for carefully revising the text. 

Cornell University, 

Ithaca, N. Y., Jan., 1893. 


iii 



PREFACE TO THE SECOND EDITION. 


On account of the increased use of transition curves for 
electric railroads, Table I, containing offsets and other transi¬ 
tion curve data, has been extended to circular curves of short 
radii. 

Table IV, supplementing Table III with actual deflections 
in degrees and minutes for a large range of transition curves, 
has been added. This table in its general form and in the 
range of curves given was suggested by the blue-print tables 
in use on the Burlington and Missouri River Railroad, and 
kindly loaned by E. E. Hart, C.E., a former resident en¬ 
gineer. When the degree of the circular curve and the length 
of the transition can be found in the table without interpola¬ 
tion, the table is more convenient than Table III. 

A five-place table of sines, cosines, tangents, and cotangents 
has been added, so that the simple curve problems involving 
transition curves can be worked in the field without the use of 
the regular field-book. 

Cornell University, 

Ithaca, N. Y., Dec., 1896. 



CONTENTS. 


SEC. PAGE 

1. Necessity for Transition Curves. 1 

2. Different Kinds in Use. 1 

THEORY. 

3. Equation of Curve. 2 

4. Coordinates of Center .. 5 

5. Deflection Angles. 5 

6. Construction of Tables. 8 

PRACTICE. 

* 

7. Field-work by Offsets. 9 

8. Field-work by Deflections. 11 

9. Best Length of Curve. 12 

10. Compound Curves. 13 

11. Tangent Distance with Transition Curve . 16 

12. Tangent Distance with Unequal Offsets. 18 

13. Transition Curve added without changing the Position 

of the Line opposite Vertex. 20 

14. Transition Curve added with the Least Deviation from 

the Old Roadbed, in improving Old Track.21 

15. Transition Curve added without changing the Length 

of the Track, to avoid cutting tbe Rails. 22 

16. Transition Curve added without disturbing the Central 

Portion of the Circular Curve.23 

17. Transition Curve added at the P.G.G. of a Compound 

Curve by changing the Degree of the Second Branch 27 

18. Transition Curve added at the P.G.G. of a Compound 

Curve by moving the P.G.G . 28 

19. Old Track.31 

20. General Case . 33 

21. Description of Right of Way.34 


v 























VI 


CONTENTS. 


PAGE 

Transition Curve Formulas for Reference.35 

Explanation of Tables... 36 

Deflections for Field Use... 39 

Table I. Offsets, etc. 41 

II. Coordinates. 57 

III. Deflection Angles. 57 

IV. Deflections in Degrees and Minutes. 59 

V. Radii for Circular Curves. 67 

VI. Arc Excess for Circular Curves. 73 

VII. Tangents and Externals fora 1° Curve. 73 

VIII. Natural Sines, Cosines, Tangents, and Cotangents 79 













TRANSITION CURVES. 


1. Necessity for Transition Curves.— 1 The road¬ 
bed upon a tangent or straight track is made level transversely; 
upon a circular curve it is inclined toward the center to coun¬ 
teract centrifugal force. The change from the one to the other 
must be gradual for easy riding, and to prevent twisting the 
trucks. Hence if the inclination of the road-bed is to be pro¬ 
portional to the centrifugal force at every point, a transition 
curve must be used, in which the curvature increases directly 
with the distance, in passing from the tangent to the circular 
curve. 

2, Different Kinds in Use.— Rankine (Civil En¬ 
gineering) gives the credit of first laying out railroad curves 
with a gradual increase of curvature, starting from the tan¬ 
gent, to Mr. Gravatt about 1828; of using such a curve to 
connect the tangent and main circular curve, to Mr. Wm. 
Froude about 1842,—their methods being first published in 
1860-1. 

In this country transition curves have come into use within 
the last ten to twelve years. The true transition curve, curva¬ 
ture increasing directly with the distance, was described in The 
Railroad Gazette, Dec. 3, 1880, by Ellis Holbrook. The Rail¬ 
road Spiral, a curve made up of circular arcs compounded, by 
W. H. Searles, was published in 1882. The cubic parabola, 
in which the curvature increases with the distance along the 
tangent, was described by C. D. Jameson and E. W. Crellin 
in the Railroad and Engineering Jour, in 1889. The Froude 
curve, as worked up by A. M. Wellington, was described in 
the Engineering News in 1890. These bring out the more 
important classes, although the list is far from complete. 

All but the last were worked out to be run with transit and 
chain, the deflections being given for several curves with 



2 


TRANSITION CURVES. 


different rates of increase of curvature for flexibility; the last 
was worked up to be run with transit, or by ordinates from the 
circular curve and from the tangent, as preferred. It is the 
only one of the list which is worked up for a curve of any 
length or of any offset from the tangent to the main circular 
curve; the field-work during location is reduced to a minimum 
by offsetting from the tangent directly to the circular curve,, 
but the fundamental assumptions become inconsistent when 
the curve includes a large central angle. 

THEORY. 

3. Equation of Curve.—In Fig. 1, let EGE' be a cir¬ 
cular curve with centre G and tangents EK and E'K. It is 



evident that if we begin at G to reduce the curvature directly 
with the distance, and continue the reduction until the curva¬ 
ture is zero, using the same central angle I as for the corre¬ 
sponding portion of the circular curve, the new curve will 
pass outside the old, requiring a tangent OB parallel with KE, 
and at some distance as JE from it. 

E is called the P. G ., as usual; 0, the P. T. G .; G, the PC,.; 
G', the P.P.C,.; E', the P.P.; O', the P.P,. 






Eq. 3.] 


EQUATION OF CURVE. 


3 


Since the curvature is zero at 0 and increases directly with 
the distance, the ratio of the curvature and distance incre¬ 
ments, d(p and dl, must increase with l , i.e.. 


§= 2 «' 


. ( 1 ) 


where 2k is a constant for any one curve, but varies for differ¬ 
ent curves. 

Integrating, 

0 = kl\ .(2) 

where 0 is the central angle up to the point L, l the length, 
and k a constant. 

In the right differential triangle. Fig. la, 
dy — dl sin 0 

= dl sin kP, since 0 = kl 2 by (2). 


The expansion of the sine in terms of the angle, by Trigo¬ 
nometry, is 

. V , t 5 # . 

sm< = < — - + - —^+, 


where ! denotes a factorial number, i.e., 31 = 1x2X3 = 6, etc. 
Substituting, 

/ m 6 kH™ kH 14 \ 

* = *(«■- 8T+1T- «-+)* 

Integrating, 

kl z m kH 11 kH™ 

y ~Y 42 + 1320 75600 *"* 9 * * # 

Restoring 0 for M 1 , 

l(p l(p z l(p 5 l<p’' . 

y “ T “ 42" + 1320 ~ 75600 '* 


where 0 is in ^r-measure, coming from the development of the 
sine in series. 







4 


TRANSITION CURVES, 


3 


For 0 in degrees ^0 = 0°~j = .O174530 9 ^, 


y = l [£(.017453)0° - ¥ V(-O17453) 3 0° 3 
H-TtW-O17453) 3 0^ - ^(.01745^0°’ -f], 

or y = 1C, ........ (4) 


where G varies with 0°; it can be taken from Table II, with 
0° as argument. 

From Fig. 1 a, 


dx = dl cos 0 

= dl cos /W 2 , since 0 = Atf 2 by (2). 

The expansion of the cosine in terms of the angle, by Trigo¬ 
nometry, is 

C0S ‘= 1-21 + 4!-6i + - 


Substituting, 


— dl (1 — — -j- 

2! ~ 4! 


W* \ 

6T~ / * 


Integrating, 


_ kH 5 kH* kH 13 
X ~ 10 + 216 “ 9360 + * * 


Restoring 0 for kP, 


x = l_ W , 10 4 _i0! , 

10 + 216 9360 




For 0 in degrees, 

a; = * — Z[ T y.O17453) 2 0 2 - ^(-O17453) 4 0 4 
4- in?W.O17453) 3 0« 

or x = l — IE, (6) 

where E varies with 0°; it can be taken from Table II with 
0° as argument. 




Eq. 11.] 


DEFLECTION ANGLES. 


5 


4. Coordinates of Center. —Since the increment to 
the central angle, dcp, is in 7r-measure, = arc dl divided by 
radius of curvature p, we have from (1), 


<U _ _ 1 

d(p~ P ’ ~ 2M' * 


. - ( 7 ) 


At theP. Ci., p — R, the radius of the circular curve; (p = I, 
the central angle of the circular curve from P.C. to P. Ci. 
Therefore, from (2) and (7), 


r_7,7 2 _ 1,1 

“2P 

There h = the length of the transition curve to P. 0\* 
But R = 5 ^; r = 7— = 57.307. 

D Tt 

= 28 - 6 4 * • • 


( 8 ) 


k = 


D 


2 Rh 11460^. 


( 9 ) 

( 10 ) 


If Xi and y x are the coordinates of P. C x and x', F, those 
of the P. C. (see Fig. 2), 


F = y x — R( 1 — cos T) 
x' = x x — R sin 1 


while the coordinates of the center will be x' and FR. 

5. Deflection Angles. —With the instrument at sta¬ 
tion 0, Fig. 2 ( P.T.C. , or P. Pi.), the tangent of the deflection 
angle d 0 can be found by dividing y by x, using the values 
from equations (3) and (5), thus: 

tan do = V - = ~ + .009 523W 
x o 

+ .000 167 kH™ +. 

* If curves from 8° to 16° are run with 50-foot chords, from 16° to 32° 
with 25-foot chords, from 32° to 80° with 10-foot chords, and all sharper 
than 80° with 5-foot chords, the above value will be correct to the nearest 
foot. The accompanying tables are computed on this basis. See p. 39. 







6 


TRANSITION CURVES. 


The expansion of the angle in terms of the tangent is, by 
Trigonometry, 

t = tan t — £ tan 3 1 £ tan 5 1 —. 

Substituting, and retaining nothing higher than k 5 

bn 

d Q = .002 823« - .000 068A^ 10 

o 

= - .002 8231% 6 - .000 068i% 10 , 

- o 

oy using for k its value 1 -f- h* from (8), and calling the ratio 
l -f- li — n. 



do and 1 are in nr-measure: to reduce to degree-measure, mul¬ 
tiply by — = -017 453, giving 


d 0 ° = —n* - .000023 22 

o 


“ - 000 000 00153 


d 0 ° = 


(i2) 









Eq. 15] 


DEFLECTION ANGLES. 


7 


i.e., the deflection varies approximately as the square of the dis¬ 
tance, and is nearly equal to one third the central angle subtend¬ 
ing the same arc. 

Bo, in thousandths of a degree, can be taken from Table III 
with 1° as argument. 

With the instrument at an intermediate point, x", y ", 


y-y" k 
x — x" 3 


tan d" = 


= o (P + r* + M") 


+* 3 [ t ^ 6 + n ++ u " 6 } 

+ ' 2 + W 4 + w 3 ) ] + & 5 [-000167(Z 10 -f- 1" M) 

+ .002 909(W" + ll"*) + .008 985(W 2 + IH" 8 ) 

+ .016 604(W" 3 + IH" ’) + .024 223 (IH"* + l 4 l" 6 ) 

+ .027 556? 5 ?" 5 ]. 

From which, by finding the value of the angle in terms of the 
tangent, and substituting the value of k as before, 

d"° = ~{P + r 2 + ir) - Correction. 

oil 

d"° = 4 (7i 2 + »" 2 4- nn") - B", .(13) 

O 

where B" contains the terms involving i 3 and I 6 , a rapidly 
converging series. 

With the instrument at Sta. 1 ( n" = 1), 


df = — (ri> +1 4- n) — Correction 

O 



3 


B: 


(14) 


With the instrument at Sta. .5 ( n " = 1), 


d°. 5 = ^-(w 2 + i 4“ i w ) — Correction 

O 



( 15 ) 





8 


TRANSITION CURVES. 


[§ 6 , 


Instrument at Sta. .25 ( n " = £), 

d \26 = jj fa 2 + iV + i») — Correction 


— 'g -4 .25 — -S.26. 


(16) 


Instrument at Sta. .75 (w" = £), 

^°.76 = ij^ 2 + + i n ) — Correction 



. ( 17 ) 


A and B can be taken from Table III, with n", n, and 7° as 
arguments. B is in thousandths of a degree. Second differ¬ 
ences should be used in interpolating for A when 7° is large. 

6. Construction of Tables.—These formulas are too 
complicated for convenient use, although all the series con¬ 
verge rapidly. Tables thus become necessary. 

Table I.—The length of curve, h , for a given B is first 
assumed in values increasing by 20 feet from 40 to 700 ft.; 7°, 
the central angle, can be found by (9); Xi , y x , thecooordinates 
of P.C.x , are found by (4) and (6) by the aid of Table II;* 
x\ F, the coordinates of the P. G. of the circular curve, are 
found by (11); l', the length of the transition curve from the 
P.T.G. to the P.G., is most conveniently found by assuming 
a value a little less than \x' -f (h - ah) + 32] for l in (5), and 
noting the resulting value of x\ a second assumption may 
sometimes be necessary to bring the value of l used in the 
higher terms to the nearest tenth, which will usually give 
V — x ' with sufficient accuracy; V — x' is tabulated as e; <p', 
the central angle corresponding with l', will = (V + by 
(2); when y' , the ordinate opposite the P.G., can be found by 
(4), or it can be found from (3) without using (p 1 ; h — V will 
give the length of the second portion of the transition curve; 
1007° “i- B -J- arc excess, taken from Table Y,f will give the 

♦They were actually computed by (3) and (5), so that occasionally 
there may be a slight discrepancy in the last place. 

+ Arc excess = excess of circular arc over the sum of the chords from 
the P.C. to the P.U.,. 






Eq. 17] 


FIELD-WOEK BY OFFSETS. 


9 


length of the circular arc for the corresponding portion; the 
difference between the transition arc and circular arc is tabu¬ 
lated as e !; e -J- e’ — L — x' — arc excess. 

Table II gives G of (4) and (B) of (6), rendering the compu¬ 
tation of the coordinates, x and y , of any point of the curve 
extremely simple. 

Table III gives A and B for computing the deflection angles 
by (12), (14), (15), (16), (17), for 20 equidistant stations on the 
curve, with the instrument at each of 5 different ones; B is in 
thousandths of a degree, and can be interpolated for interme¬ 
diate values of (p and I. A should be interpolated by second 
differences when 1 is large, or directly computed if more con¬ 
venient. As great flexibility is thus given in the field-work 
in passing obstructions, etc., as with circular curves. 

All the tabular quantities were computed to one or more 
places than have been retained, and they have been checked, in 
type, by second differences. It is confidently believed that no 
errors of importance will be discovered, and no inaccuracies 
which can affect the best grade of field-work. 

Linear interpolation will usually be sufficient for interme¬ 
diate values; second differences may be preferred for F and y 
in some parts of the table. 

®i» V\> will usually be of use when an obstruction, or other 
local condition, prevents the use of the ordinary methods in 
the field. 

Table IV.—For convenience, when the degree of the circu¬ 
lar curve can be taken as a whole degree and the length of the 
transition curve can be slightly modified to suit the table, 
Table IV has been added giving the actual deflections in de¬ 
grees and minutes required in running the curve. It is com¬ 
puted from Table III, but for fixed chord lengths rather than 
for a fixed number of chords, as being more convenient for 
tabulation and for use. The offsets Fhave been added from 
Table I, and the long chords from computation. 

PRACTICE. 

7. Field-work by Offsets. —It will usually be most 
convenient during location and construction to simply allow 
the offset F between the tangent and circular curve, and the 


10 


TRANSITION CURVES. 


excess in length when important, without running the transi¬ 
tion curves Two hubs are then required at the P.G., and also 
at the P.T., one at the P. G. or P.T. proper, and the other on 
the tangent opposite, the distance between being F, one being 
set as a foresight and the other by offset. The instrument 
when brought up is placed over the offset hub, requiring the 
backsight to be similarly offsetted ; or, with a table of natural 
functions, the vernier may be set to 

cot -1 (distance -f- offset); 

or, if the offset is small, at the angle in minutes 

_ offset 

~ .029 X distance in stations' 

On a curve the offset should be multiplied by cos (nD 2),* 
and the long chord taken for distance, in the above expression, 
when failing to offset the back flag. 

The excess in length, e+e', must be added to the station 
numbering, making the change for convenience at the P.C. 
and at the P.T. the same as the deficiency in length of the 
circle as compared with the sum of the tangents must be sub¬ 
tracted from the station numbering of the P.T. when the 
latter is set from the P.I. and the tangent continued without 
running the circular curve. 

If intermediate points on the transition curve are needed 
during construction measure ordinates from the tangent pro¬ 
portional to the cubes of the distances from the P.T.C., or 
from the circle proportional to the cubes of the distances from 
the P.C. i, F being about double the ordinate either way at 
the P.C. 

Example 1. At station 674 a 12° curve to the left begins. Required 
the data for a 500' transition curve (front tangent not fixed). 

From Table I for Z, = 500, F — 21.GO, e = .4, e’ = 1.8, x' = 247.7. 

at sta. 674, -~-.(e -f- e') = 673 -(- 97,8, but called 674 to allow for excess 
in length, erect a perpendicular with transit or chain = 21.60; place hub; 
set transit over it with 0° of vernier parallel to tangent and run curve 
as usual. The P.T.O. will be 673 -j- 97.8 - a?' = 671 -f 50, and the P.C.^, 
676 + 50. 


* n = distance in stations; D = degree of curve. 






Eq. 17.] FIELD-WORK BY DEFLECTIONS. 


11 


Sta. 672, 


672 + 50, 


?i^ + 53 = .0864; 
2 

.0864 X 23 = .7; 


673 .0864 X 33 = 2.3; 

673 + 50, .0864 X 43 = 5.5; 


and the same values for corresponding stations from the circle. 

The true coordinates for any point can be computed by (4) and (6), if 
needed for greater accuracy. Thus, for sta. 673: Table I gives J= 30°, 
(2) gives <f>° = 30* = 2°.7. Table II gives C = .0157, E = .00022. 

.-. y = 150 C = 2.36; x = 150 - 150 E = 150. 

8. Field-work by Deflections.— For track centers 
the transition curve is best run with a transit. From F, I and 
h are readily found from Table I, interpolating if necessary. 
Having h, the number of stations (an aliquot part of 20) can 
be taken so that they will be no farther apart than on the cir¬ 
cular curve. The instrument points for running the curve are 
then decided upon, the corresponding numbers are taken from 
Table III to multiply by in- 3 for the deflections, subtracting 
the corrections when important. The curve is then run in the 
usual way with transit and chain, but with short chords. 

Blank pages are inserted after page 40 in order to preserve 
the deflections computed for the more common transition 
curves. 

Regular stations can be used if preferred, it adding slightly 
to the labor of finding A in the deflection formula. 

When the curve can be run with the instrument at theP T.C-, 
—and 1 0 is not over 15° to 20°, the deflections can be taken by 

(12) equal without the use of Table III. 

U 

When the degree of the curve and the offset can be taken 
from Table IV without interpolation, it will be found more 
convenient as the deflections can be taken out directly without 
computation. 

Strictly, the distances measured should be along the arcs, 
and not along the chords; the chords can be shortened by 
Table VI to allow for this whenever the accuracy of the chain¬ 
ing will warrant; the maximum discrepancy is one inch per 
100 feet, if the chords are as short as would be used for circu¬ 
lar arcs of the same curvature. 


12 


TRANSITION CURVES. 


[§ 9 , 

In plotting, the radius of the circle is increased by the offset 
in fixing the center from the tangent, while the radius is used 
in describing the arc, thus leaving the offset F at the tangent 
point. 

Example 1. —Find the deflections for a 9° curve with a 10.43-ft. offset. 

Table I: 7=18°; 7-j-3 = 6°; h = 400'; requiring 10 chords each 40'. 

With instrument at P.T.C., from Table III: 

do-^.OlXfl^O 0 3'.6; do- 8 = .64X6° =3°50'.4; 

do. 2 = .04X6 =0 14 .4; do- 9 = .81X6 -.003 = 4 51 .4; 
. do 1 =1 X6 -.005 = 5 59 .7. 

On reaching sta. 1 or the P.C. U set up, backsight on sta. O, deflect 
18° — do 1 f° r tangent at sta. 1; then run the remainder of the circular 
curve as usual. On reaching the P.T.C. j, the second transition curve 
can be run continuously by placing the circle so that the vernier shall 
read 1° for the tangent at P.T.C. x , and zero when parallel with the tan¬ 
gent at the P.T., and then setting the following readings computed from 
Table III.: 

Instrument at P. T. C. i: 

di. 9 = 2.71X6° = 16° 15'.6; di. 2 =1.24X6°-.003 = 7° 26'.2; 
d 1 - 8 = 2.44X6 =14 38.4; di.i=l.llX6 -.004 = 6 39.4; 

. di° =1 X6 -.005-5 59 .7. 

Example 2.—Find the deflections for a 9° curve with an 8.45 ft. offset. 

Table IV : 40-ft. chords; 7 = 16° 12'; l t = 360, or 9 40-ft. chords. 

With instrument at P.T.C. , or sta. O : 

Deflection for chord point 2 = 0° 16' 

“3=0 36 
“ “ “ “4=1 04 


“ “ “ “ 9 = 5 23 .8 

Tangent at “ “ 9 = 16 12 

The deflections are also given for the instrument at chord point 2, 80 
feet from the P.T.C .; chord point 4; chord point 6; chord point 8; and 
chord point 9, the P.C 2 . 

9. Best Length of Curve.— The rate of change of 
elevation of outer rail preferred by engineers, in passing from 
the tangent to the circular curve, varies from 1 to 4 inches per 
station; the elevation on the circular curve varies from one- 





Eq. 17.] 


COMPOUND CURVES. 


13 


half an inch for low speeds, to 4 inches or more for very high 
speeds, per degree, until a maximum of from 5 to 9 inches 
is reached. This would require from 10 to 200 feet of tran¬ 
sition curve per degree of main curve up to a length of some 
600 feet. 

On short tangents and short curves short transition curves 
will be required to prevent serious overlapping. On the other 
hand, a circular curve with transition ends leading to the tan- 
gents will usually approach nearer to the shape of a contour 
around a spur or up a narrow valley than a circular curve with 
tangents, thus using the longer curves in extending the com¬ 
binations available for fitting the ground in rugged topography. 
This flexibility will also be found very convenient in running 
track centers over crooked portions of the line, as the line can 
be better fitted to the road-bed, and small discrepancies in re¬ 
running the curves from imperfectly recovered points can be 
taken care of in the offset, as any offset within the limits of 
the tables can be used by interpolation. 

lO. Compound Curves.—The general relation of a 
transition curve to a compound curve can be seen by refer¬ 
ence to Table IV, which gives the curvature at each chord 
point of the transition curve and shows that the portion of 
the curve between any two chord points would form a 
transition curve connecting the two circular curves the 
same as the portion from the zero end or P.T.G. to any 
chord point forms a transition curve from the straight line 
to the curve. 

To find the data required for locating the tangent points, as 
a check upon the transition curve, in Fig, 3, let G be the cen¬ 
ter of the sharper curve of radius R and degree D; G ", that of 
the flatter curve of radius R" and degree D" ; F it the perpen¬ 
dicular distance E'J' between the curves on the line joining 
their centers; OG"G, the transition curve, joining the first 
curve at G, the second at G", and its own tangent OB (not the 
tangent to the compound curve) at O ; h , the length OG; 
l ", the length OG" ; l- 2 , the length G"G ; e 2 , the excess in 
length by using the transition curve. Call G the P.T.C.i\ 
E' or J\ the P.C.G. ; G" the P.C . 3 ; E" the P.T." ; and O 
the P.T. a. 


14 


TRAKSITIOK CURVES. 


[§ 10, 


Since the curvature increases directly with the distance, 

) 


" = 4 = 


4 =h -1" = h - ~ U I> ‘ . j 


. . . . ( 18 ) 



The coordinates of the center Uare x' and FR, taken from Table I 
for the D° curve ; those for C"'are*' 2 and F" 4- R ", taken from Table I 
for the D" curve with l" as argument. 















Eq. 20.] 


COMPOUND CURVES. 


15 


In the right triangle HCC", HC, and HC" are known from the above, 
giving 

t _ HC _ x ' — x \ _. ~ 

~ HC"~~ F" 4- R" -F-R' 


CC" = — — . = V HC 2 + DC'" 2 ; 
cos i 


F 2 = R" - R - CC "; 

GE', or P.r.C.a to P.C.C. = 1 -jym; 

J'G", or P.C.C. toP.C. 2 = ^^100; 

e 3 ==-- Z 3 — ( GE" + J'G") — arc excess; 

= R — 100 /-— - + i— — arc excess; 
\ D D" I 


• (20) 


where “ arc excess ” is the sum taken from Table VI for the proper 
lengths for the two circular arcs. 

Computing P 2 ancl e z< f° r different cases, they are found to be the same 
as given by Table I, with D-D" and Z 2 as arguments, up to about 
Dl x = 8000 ; i.e., up to D= :2°, - 680' ; D = 20°, Z, = 400' ; etc. For 

greater values, P 2 should be computed by (20) for accuracy, although 
the error is not large at the limits of the Table. 

Computing GE', the distance from the P.T.C.% to the P.C.C. on the 
sharper curve, by (20), it is found equal to Z 2 -r-2, up to about PZ, = 4000; 
leaving J'G", the distance from thePU.C. to the P.C. 2 on the flatter 
curve, = Z 2 -s- 2 — e. z — arc excess. For much larger values these distances 
should be computed by (20). 

Hence, in using the offset method of allowing for the transi¬ 
tion curve, assume a suitable U (or F 2 ) for a D — D" curve; 
take out the corresponding ^(or h) and e-\-e’ from Table I; 
run the circular curve to E', the P.G.G., as usual (the front 
curve not being closely fixed in position); offset radially an 
amount F 2 \ set up at J and continue as usual, allowing for 
e + e when important. 

In using the deflection method, assume the length h be¬ 
tween the two branches from Table IV, and take out the 
length of the transition curve to each point, h and l". From 
Table I take out the offset F a and the excess length e 2 (= e -f- 
e') for a D — B" curve. Then GE', the distance from the 
P.G.G. to the tangent point of the sharper curve = 1% ■+■ 2 ; 
J'G", the distance to the tangent point of the flatter curve, = 
l % -f- 2 — — arc excess for both circular curves taken from 

Table YI. Lay off GE' and J'G", locating the tangent points 










16 


TRANSITION CURVES. 


P.T.C. a and P.C. a ; then set up at either one and run the 
curve from the deflections given in Table IV. 

If Dh exceeds the limits given above. F a , e a , GE' and J'G " 
must be computed by (20) as already indicated. 

If the length of the transition curve cannot be assumed 
from Table IV, it can be assumed to correspond with Table I, 
and F-x and e a taken out as above. The deflections will then 
have to be computed from Tuble III, making it more con¬ 
venient to run the curve from the P.T.C. 2 , the tangent point 
with the sharp curve, unless the P. C. a should happen to come 
at an instrument point of the table. 

Example 1.—A 10° curve compounds to a 4° with a 3.92 ft. offset, at 
sta. 196 + 40. Find the data for running the transition curve with a 
transit. 

Table I, for F a = 3.92, and D - D" = 6°, gives Z 2 = 300 ft. 

By (18), Z, = Z 2 jy ^ Dn = 500 ft.; l" = l x - 1 2 = 200 ft. 


Table I, for D - 10°, Z, = 500, gives 1= 25°, F = 18.05, x' = 248.4; for 
D" — 4°, l" = 200, gives I" = 4°, F" = 1.16, x\ = 100. 

Table V gives R = 573, R" = 1432.5. 

DZ I > 4000, requiring a computation for GE' by (20) for strict accuracy. 

tan Z = —_ g 48 ' 4 - 100 - 

1.16 + 1432.5 - 18.05 - 573 842.61 ’ 

giving i — 9°. 989. 


GE' = 
J'G" = 


25 - 9.989 
10 

9.989- 4 


100 = 150.11 ft. 


100 = 149.72 ft. 


P.T.C.z = sta. 196 + 40 -(1 + 50.1)= 194 + 89.9; 

P C . a = “ 194+ 89.9+ (3 + 00) = 197 f- 89.9. 

Table IV, instrument at P.T.C. 2 , or chord point 10. 

Tangent at C.P. 10 = 25° 00 7 

Deflection for “ 9 = 22 35 

“ “ “ 8 = 20 20 

“ “ “ 4 = 12 59.8 

Tangent at “ 4 = 4 00 

Zero at P.T.C. 2 makes an angle = 25° with tangent to 10° curve. 
Tangent at P.C. a makes with backsight an angle= 12°59 , .8-4° = 8° 59'.8. 
In this example the deflections could have been taken from Table III. 
with nearly the same readiness. 

11. Tangent Distance witli Transition Curve. 

—Given the central angle A, the degree of the curve D, and 
the offset F, to find the tangent distance V. 







Eq. 23.] 


TANGENT DISTANCE. 


17 


The effect of the transition curve is to move the circular 
curve from the position HLM to ENE ', the perpendicular dis¬ 



tance between the tangents being F. In the right triangle 
JCB, 


JB = (EC + JE) tan^, 
2 


or 

T' = (R + F) tan |.(21) 

= T+F tau^. ..(22) 

2 


T, = B tan — , being the tangent distance without the transi- 

tion curve, found by Table VII, or by computation. 

If the curve is to be run by deflections, x' must be added to 
T' to reach back to the P.T.G., giving 

Ti — T -\- F tan ^ -f x\ .... (23) 


x' can be taken from Table I with D and F as arguments. 

Example 1.—Write out the transit notes for a 6° curve, with P.I., point 
of intersection of the two tangents, = sta. 721 + 52.7; A = 17° 21'; F = 
2.51 ft. 









18 


TRANSITION CURVES, 


[§ 12 , 

By (21), T = (955 -f 2.51) tan 8° 40*' = 146.1. 

A 

Length of curve = — = 2.892 stas. 


TRANSIT NOTES. 


Sta. 

Point. 

Curve or Bearing. 

Vernier. 

-f 95.8 

&T.,F=2 .51 

S. 35° 30' W. 

8° 40*' 

2 



5 48 

1 



2 48 

-f 6.6 

P.C., F = 2.51 

6° R. 


720 


S. 18° W. 



For track centers, or transition curve run with transit: 
By (23) and Table I, T, = 146.1 + 119.9 = 266. 1° = 7°SI. 

A — 21 

Length of circular curve = —-— = 0.492 stas. 


Length of transition curve, Table I, = 240 ft., requiring 4 60-ft. chords. 
Table III, for I = 7°.2 : 

Instrument at sta. O: d 0 - 25 = 0° 09'; d„- 8 = 0° 36'; d 0 - 75 = 1° 21'; do 1 = 
2* 24'. 

Instrument at sta. 1: d,i = 7° 12'; d,- 78 = 5 e 33'; d ,•» = 4° 12'; d,- 28 = 
3° 09'; d,° = 2° 24'. 

Table IV, 40-ft. chords would give the deflections also, but for 6 chords 


instead of 4. 


TRANSIT 

NOTES, 

• 

Sta. 

Point. 

Curve or VprTlipr 
Bearing. vernier. 

Remarks. 

724 +15.9© 

p.r. i 

S. 35° 30' W. 

2° 24' 

Vernier = 2° 24' for B. S. 

.25 



3 09 


.5 



4 12 


.75 



5 33 


4-75.9© 

P.T.C.j 


1 28* 

Vernier = 8° 404' for B. S. 





(= 1° -f 1° 284') 

721 4-26.7© 

PC., 

6* R. 

2 24 

Vernier = 4° 48' for B. S. 





(= 1° - 2° 24') 

... ,75 



1 21 

(These settings for B. S. 

.5 



0 36 

give 0° on the tang, for 

.25 



0 09 

P.C., and P.T.j, and 1° on 

+ 86.7© 

P.T.C. 



the tang, for P.T.C.,.) 

718 


S. 18° W. 




12. Tangent Distance witli Unequal Offsets.— 
From Fig 5, 


T - T+AH- BJ 







Eq. 25.] 


TANGENT DISTANCE. 


19- 


T" = T+ BII — GH 

F' F" 

- T4- —- 

sin z/ tan A 




A F' F" 
= Ii (an - + -4-; - 

2 sm A tan A 


(25) 



Example 1. Compute the data for a 4° curve; P.I. = sta. 54 -+• 35.6; 
4 = 23° 15'; F* = 2.62 ft.; F" = 0.94 ft 


By (24), 


V = 294.7 


0.94 2.62 

.3947 .4296 


291.0. 


(25), 


r£if 


294.7 + 


2.62 

.3947 


0.94 

.4296 


299.2. 


Length of curve = — = 5 812 stas. 

.*. P C. = 51 + 44.6, with offset F f = 2.62 

P.T. = 57 + 25.8, with offset F" = 0.94. 

No further attention need be given the transition curves, except to 
allow the proper offsets until track centers are required. 

Ex. 2. Write out the transit notes for Ex. 1 for track centers. 

















20 


TRANSITION CURVES. 


[§ 13 , 

13. Transition Curve added without chang¬ 
ing-the Position of the Line opposite Vertex.— 

Let E be the external for the old curve, and E' that for the 
new. In Fig. 6, 

PF=BN-BP, 


E' = E — Faec —. 

2 


. (26) 



Divide the external distance for a 1° curve. Table VII, by I) 

for E, or compute it from E = R ^sec ~ — lj. Then find E' 

by (26), and divide the external for a 1° curve by it for B ', the 
degree of the curve required. 

D' can always be changed to the nearest whole minute for 
convenience in running without sensibly disturbing the re¬ 
quired position. 

It will be larger than D. 

Example 1. A 4° curve beginning at sta. 80 and ending at sta. 85 is 
to be replaced by a curve with a 1-ft. offset passing through the same 
point opposite the vertex. 












Eq. 27.] 


FIELD PROBLEMS. 


21 


Required the field-notes running the entire curves with a transit and 
using regular stas. 

Table VII: for A = 20°, external for 1° curve = 88.40, tang* nt = 1010.4. 
E - 22.10, T = 252.6. 

E' = 22.10 ~ = 21 08; 

D' = 4° 11'.6, say 4° 12'; 

<Z 1 .Uv) 

T = -^^+ .2 = 240.8. 

4.2 

Table I: 

for = 180, D = 4? 13*; F= .94 -f J§(U8 - .94) ^ 0.99; 
h = 200, D 4° 12'; F = 1.16 + £§+45 - 1.16) = 1.22. 

1 _ 0 99 

.-. for F = 1, D = 4° 12'; Z, = 180 + 20-- ^ ¥ 180 9 , x> = 90 4 - 

By (9), /° = M = 3“.799. 

20 — 2 X 3 799 

Length of circular curve =- ; = 2 953 stas. 

4.2 

P.T.C. = sta 80 + T - V - x' = 79 + 21 4. 

The eTs are interpolated from Table III. 


TRANSIT NOTES. 


Sta 

Point. 

Vernier. 

Remarks. 

4* 78.5 C 

P-r 

1° 

16' 

Vernier = 1° 16' for backsight. 

S 

A 


2 

3.3 


4 

+ 97 60 

r.T.c.j 

6 

12 

Vernier =^10° for backsight 

3 


4 

9 

(= P + 6° 12'). 

O 


2 

3 


+ 2.30 

P-C \ 

1 

16 

Vernier = 2° 32' for backsight 

1 


1 

13.8 

(= P - 1° 16'). 

80 


0 

14.4 


79 + 21.40 

P.T.C. 





14. Transition Curve added with the Least 
Deviation from the Old Roadbed, in improving 
Old Track. —For this case the new track should pass about 
as far outside opposite the vertex as it passes inside opposite 
the P. G., or about F -s- 2. 

A F 

.’.E f =E-F sec|~g ; * * 


. . ( 27 ) 





TRANSITION CURVES. 


22 


the problem otherwise being the same as § 13. Instead of 
F 2, any other value can be used as desired. 

15. Transition Curve added without chang¬ 
ing the Length of the Track, to avoid cutting 
the Kails.— In Fig. 7, let A and H be the P.O. andP.P. for 


o • 



; 0 ' 


the old track; G and G' the P.G. and P.T., 0 and 0' the 
P.T.G. and P.T. X , for the new track. 

By the old line, 

Dist. from 0 to 0' = OA -j- arc AH -f HO' 


= 20A -f PA 
= 2(T'+x' - T) + PA 

= 2 ^(P' -f F) tan ^ -f x — 



where A is in ar-measure = .017453 A°. 
By the new line. 


Dist. from 0 to O' = 2(x' -J- e -|- e') -f arc GG' 
, = 2(x -}- e -f- e') -j- P A ; 

where x' and e-j-tf are taken from Table I. 
















Eq. 29.] FIELD PROBLEMS. 

Placing tbe two values equal, 


23 


JR' A + 2{x' + e + e') 

[ A A\ 

= 2f (R' + F) tan - -f x' - R tan - j + RA\ 


.017457^7° - 2(R - F) tan -- 2(e + e') 

*'=- -- : -4-• . (28) 

.01745^° - 2 tan -- 


or 

.008725(12' - R)A° + (e + e') + (R - R') tan 4 
F= - 2 (29) 

tan 2 

This problem is usually limited to small values of. F or to 
small changes in R, so that e -f- e' is practically known. For 
larger changes a second trial may be necessary. 


Example 1. Track has been laid on a 5° curve 900' long. Required the 
data for transition curves without changing the length of the rails. 
Assume F — 1.5 ft., as it will give by § 9 an easy transition without too 
great shift of track. 

Table I; e -f e' = 0. Also, A = 45°; R = 5730 -s- 5 = 1146. 

.' • ' ,J 


(28), R' = 


899.90 - 948.13 - .00 
.78525 - .82842 


= 1117 ft. 


Changing R' to 1116.2 ft., 

D' = 5° 08'; L' = 45 -h 5.133 = 8.767 stas.; V — (1116.2 +1.5) tan % = 463.0; 


T for old curve = 474.7, requiring the P.C. to be moved forward 11.7 ft. 

Table I; by interpolation Z x = 200.7 ft. 

With stakes 100' apart up to 8° curves, it is unnecessary to run the 
transition curve with an instrument; y'(= P-s-2 in this case) locating a 
point opposite P.C.; x’ measured back on the tangent locating P.T.O., 
and I, = ZjP-s-200, locating the P.C ,, each being distant Z, -s-2 from the 
P.C. in this case. 

16. Transition Curve added without disturb¬ 
ing' the Central Portion of the Circular Curve. 

-r-To make room for a given offset F, the end of the circular 
curve must be sharpened by compounding, and to prevent too 
great a difference in curvature at the P.G.C., the change 
should be limited to, say, 1° to 2°, i.e., D’° — D° < 2°. 





24 


TRAJSTSITIOK curves. 


The two circular curves are tangent to each other at G, 
Fig. 8, and the angle between them increases directly with the 
distauce. Hence the offset from one to the other must increase 
with the square of the distance so long as the angle can be 
taken proportional to its sine, and the distance can be taken 
the same by either curve. 

At 100 feet from G, 

Offset = 100 sin •£(/>' — D) = 50 sin 1° ( H° _ Z>°) 

= f 

Hence at n stations from G, 


Offset = fw 2 (Z) /0 — P 0 ),* approximately, or, ) 

F < fft/ 2 , for (Z>'° — JD°) <2°. f * ^ 

From this approximate value of n assume a convenient A x . 
Then in Fig. 8, 

BE- AG-AH- CE, 


or 

from which 


R = R-F-(R- R') cos A Xi 



F 

cosAx 


(31) 


Having R ', the transition curve can be put in as usual. 

For flat curves and short transition curves the problem can 
be simplified as follows : 

Assume the P. G. x from 100 to 200 feet from the P. G .; meas¬ 
ure the perpendicular distance y x to the tangent, and the angle 
/which the curve at the P. G. x makes with the tangent. If 
the curve is regular and the P G. can be accurately found, y x 
and I can be computed from AG or n. The coefficients G and 
Eoi equations (4) and (6) can be found from Table II with 1 
as argument, giving 


l - y% - 
ll ~G' 


(32) 


Xi = h — l x E. 


(33) 


* This formula is very convenient in location, in computing the change 
of curvature required for a given offset. When D = 0, it reduces to 

F — |n 2 X>, 

the offset from a tangent to a circular curve. 







Eq. 33.] 


FIELD PROBLEMS. 


25 


Measure x x back on the tangent from the foot of the perpen¬ 
dicular at P. (7.i for the P.T.C. One or two intermediate 
points can be located by coordinates computed by Table II; 
or the curve can be run with transit, computing the deflection 
angles by Table III, or taking them proportional to the square 
of the distance with the instrument at the P.T. G., the last one 
being I- 5-3, which will give a check on the work. 



An approximate value of h can be found from (32) by sub 
stituting the first term of the equation above (4) for G, giving 

l = 

1 .0174537° * 


By the footnote to (30), 

y x = |?i 2 7>. 






26 


TRANSITION CURVES. 


i 


Substituting, 

h = 150» ! p, 
or 

h = W0n; .(34) 

i.e., the length of the transition curve is once and a half that of 
the circular curve from the P. G. to the P.G.i. 

The degree of curvature at the P. G.i as found by (9) will be 

TV 1 2 ™L° . 

h 9 

by (34), 

_ 2007 * _ 200 
“ 150 n ~ 150' * 
or 

& = & .(35) 

The method is thus admissible to 3° curves if the break in 
curvature is limited to 1°, to 6° curves if the break is limited 
to 2°, etc., independently of the distance from the P.G. to the 
P.G.i, or of the amount of the old track disturbed. 

It can be extended to the case where there is an offset be¬ 
tween the tangent and the circular curve at the P. G., if desired; 
although it then offers no advantages over the regular methods 
already given, while it has the disadvantage of usually intro¬ 
ducing a break in curvature at the P.G.i. As the offset at the 
P. G. increases relatively to the offset y x at the P.G.i , the break 
in curvature decreases, reaching zero when the ratio becomes 
1 : 4; the transition curve having the flatter curvature (D ' < D) 
at the P.G.i for all greater ratios. 

Example 1. Find the data for a 4-ft. offset for a long 12° curve with¬ 
out disturbing the central portion. 

(30), 4 < In 2 , giving n > 1.51. 

Assume Aj = 24°, giving 

(3!). «'-4W.B- T -i^ 5 = «lA 

Table V, D 1 = 13° 17'. 

The P.C. will be moved forward, = 200 — ^>100 = 19.3 ft. 





Eq. 36.] 


FIELD PROBLEMS. 


27 


Table I, with F = 4 ft., D’ = 13® 17'; ?! = 203.7; = 101.6; 1= ^ = 

200 

13°.529; P.C. to P.C. X = l x + 2 = 101.8. 

The deflections can now be found from Table III, or the x and y Co- 
Ordinates from Table II. 

By the second method; assume the P.C. X , say 100 feet on the curve; 
the P.T G. by (34) will be 50 feet back on the tangent; and the middle 
ordinate will be ^ that at P.C. X \ the latter being readily measured on the 
ground, or it can be computed. The break in curvature at the P.C . X , 
however, will be by (35) D’ — D =.-1 x 12 — 12 = 4°, as compared with 1° 
17' at the P.C.C. by the first method. The former cannot be reduced, 
while the latter can be by diminishing F, or by increasing the distance 
from the P.C. to the P.C.C. 

17. Transition Curve added at the P.C.C, of 
a Compound Curve by changing the Degree of 
the Second Branch.— Draw a tangent AB = T at the 
P.C.C ., Fig. 9, to meet the tangent at the P.T. at B, and meaa 
nre AB and A. 



Let the required tangent he T. 

In the right triangles BJII and BEG, 

GE=BH=F cot A. 

.*. T' = T± Fcot A; .(86) 

where the plus sign is for the second branch sharper than the 
first, and the minus sign for the second branch flatter than the 
first. 

To find D’, divide the tangent for a 1° curve, Table VII, by 
F, or 






28 


TRANSITION CURVES. 


R = r cot 

a 


jy — 


5730 
R ' * 


Having D\ proceed as usual. 

If an offset F' is added at the P. T. also, T’ will be shortened 
by F' -t- sin A , giving 

T" = T ± Foot A - * .(37) 

sin A v ' 


with which proceed as above. 

An offset is readily added at the P. G. if desired, from what 
has already been given. 

Example 1. A 20° curve compounds to a 12° at sta. 844, with the P.T. 
at sta. 850. 

Required the data for a 3-ft. offset at the P.C.C. and a 7-ft. offset at 
the P.T. 

A = 72°; Fz=S; F> = 7; T = R tan ~ = 346.9. 


By (37), T" = 346.9 - 3 X .325 - = 338.5. 


Z>" = 


5730 

338.5 X 1.37638 


= 12°.3 = 12° 18'. 


The new vertex comes opposite a point on the tangent back of the old by 
an amount 


F 

sin A 


+ F' cot A 


= “fel +7x .325 = 5.4 ft., 

which places the new P.T. back of the old an amount 
= 5.4 + (346.9 - 338.5) = 13.8 ft. 


18. Transition Curve added at the P.C.C. of 
a Compound Curve by moving- tlie P.C.C. 

Let Ri = the radius and G the center of Ihe first branch; 
= the radius and 67 the center of the second branch; A\ = 
the central angle of the second branch; A\ — the required 
central angle with the offset F added at the P.C.G.; A’\ = 
the required central angle with the offset F added at the P. G. G. 
and Fi at the P. T. 

(a) i? 2 < Ri (Fig. 10). 

With G as center and Ri — i? 2 — F as radius, describe the 
arc GG' to meet a parallel to the tangent at P.T. drawn 




Eq. 40.] 


FIELD PROBLEMS. 


29 


through the center Ci. G will be the new center required, 
and J the new P. 0. G. 

In the right triangle OCiK, 

CK = CCi cos A<i = (Hi — R 2 ) cos 
In the right triangle CGK, 


cos A\ — 


CK 
GG * 


cos A\ 


(Ri — I? 2 ) cos A 2 
i?i — i?2 — F 


(38) 


A B cl 



With an offset F 2 at the P.T ., G' will be F% farther from 
AB than G, giving 


CK' 

cosJ , = ^r; 


COS A" a = 


(Bi — i? 2 ) cos At — F<t 

Rt — K — F 


0 0 9 0 


(39) 


(b). Ri > Ri. (Fig. 11.) 

CK = (R 2 — Ri) cos A 2 ; 


COS A 'a 


(R 2 — Ri) cos A 2 
Ri — Ri — F 


(40) 














30 


TRANSITION CURVES. 


Or, with offset F 2 at P. T., 

„„„ A K _ ~~ C0S ^2 “f- Fl 

COS Z/q < 

Ii2 - 


( 41 ) 


H 

A 0-' B 



Example 1. A 6° curve begins at sta. 672, compounds to an 18° at sta. 
680, with a tangent at sta. 685. Required the data for an offset of 5.01 
ft. at the P.C.C., and 32 ft. at the P.T. 


A 2 = 90°; F = 5 01; F u = 32 ft. 


(39), 


COB A", — (9> 55 - 318.33)0 - 32 . 
3 955- 318.33 -5.01 ’ 


giving A" a = 92° 54'. 


P.C.C. = 680 + - a n A " a = 679 + 51,7? 

u x 

P.T. = 679 + 51.7 + 7^- + (e + O = 684 -f 68.0; 

taking e + e’ from Table I, for D = 12°, F = 5.01. 

The deflections for the transition curve with the instrument at the 
P. T. C. would be computed as follows, using the notation of § 10 and Fig. 3. 













Eq. 41.] OLD TRACK. 31 

Table I, for D - D" = 12°, and F = 5.01, gives 7, = 240. 
lj = 240 X If = 360, requiring 10 36-ft. chords; l" - 360 - 240 = 120. 

D = 18°, l x = 360, gives 1= 32°.4, x ' = 178.1, F+R = 16.76 + 318.33 = 335.09. 
£>" = 6°, l " = 120, gives I" = 3°.6, x" = 60, F" + R" = .63 -f 955 = 955.63. 

«*»• ta “ 4 = 95 o!m-~3 |- 09 : * = 10 “ 46 '‘ ; 

G"J' = P.C. 2 to P.O.C. = ^£'l00 = 119.6; 

E"G" = P.T." to P.C .2 = ^ilOO = 60; 

PG = P.C.C. to P.P.C.a = —100 = 120.1. 

Table III. do 1 = .01 X 10°.8 = 0° 6^'; 


d 0 a = i X 10.8 = 1 12 = p.a a ; 


do 1 = 1 X 10.8 - .030 = 10 46 = P.P.C. 3 . 

.-. set up at E" at sta. 679 + 51.7 - 119.6 - 60 = 677 + 72.1, turn tan¬ 
gent; measure x" — 60 back; offset F" = .63, giving O; set up at O and 
run transition curve with above deflections and 36-ft. chords, except 
when interpolating d Q h. 

Table IV, 20-ft. chords would give the deflections without computation, 
and allow of setting up at the P.C. a as well as at the P.T.G . 2 . 

19. Old Track.— The problems already given will suffice 
for all the ordinary cases in aligning old track, where the 
curves are short and the tangents can be run to intersect at 
the P.I., or can be connected across by a single tie-line. 

With long sharp curves, where the roadbed usually lies in a 
deep cut or high fill, so that it is inconvenient to prolong the 
tangents to intersect, other methods will be found more de¬ 
sirable. On account of the trackman’s habit of springing the 
tangent points in, making a sort of transition curve in his 
effort to secure an easy-riding track, the degree of curve can 
best be found by setting up on the central portion, and noting 
the deflections per station each way from the instrument. A 
value can then be assumed, and a trial-line extended through 
in each direction, noting distances from the center of the track 







TRANSITION CURVES. 


32 


[§ 19 , 


at different points, and especially at the ends, in order to de¬ 
termine the offsets to the tangents. 

The change in deflection, when small, required to pass the 
line through any given point can be found from (30), 

Offset = | n\B' - D) = \n\d' -d);* 
where n = distance in stations, B and d the trial degree of curve 
and deflection, respectively, and D' and d f the required ones. 

The change in the position of the first instrument point with 
reference to the center line, and the deflections which will best 
fit the roadbed, can thus be readily found. Slight changes in 
deflection, starting at instrument points, are admissible in 
order to lessen the shift of the track, the effect being to slightly 
compound the curve ; while any slight discrepancy in the 
work can be thrown into the offset at the tangent and taken 
care of in the transition curve. 

If in thus fitting the roadbed a suitable offset cannot be 
secured for the transition curve without too great expense for 
track-work, the end may be compounded by one of the methods 
of § 16. If the track is shifted more than, say, 2 feet, the 
estimated cost of track-work should include the extra cost of 
maintenance, due to the settlement of the ends of the ties on 
the new roadbed. 

If the original curve is compound, the P. G. G. will be diffi¬ 
cult to find directly with accuracy, owing to the approximate 
transition curve introduced by the trackman. In running the 
tiial-line which fits the first branch of the compound curve, 
the distances from the center of the track will begin to increase 
in passing onto the second branch. Measuring two of these 
and substituting in (30), 

Offset = $n\B' - B ); 

Offset'= !(n + l)2(P'_P) ; 

from which to find n andP', the other quantities being known. 
Or, B can be found directly in the same manner as B, when 
one measured offset will give n, or the distance from the offset 
to the P. G. G. 

Having the P.G.G., if the change in curvature is great a 
transition curve can be added by the methods given in §§ 17, 
18. Or a short transition curve ca n be added without chang- 

* Searles’s Field Engineering , Table X and §§ 146-152, on the valvoid 
curve, will give more accurate results, but with more labor. 





Eq. 41.] 


GENERAL CASE. 


33 


ing the main portion of either branch of the compound curve 
by sharpening the sharp curve, and flattening the flat one by 
say 1°, for a distance n from the P.G.C ., so that the two 
changes will give the desired offset by (30). The curvature 
on one side will thus be increased as much as it is flattened on 
Ihe other, leaving the two branches parallel at the P. G.C., and 
at the offset distance apart. The transition curve can then be 
run by § 10; or, with a close approximation for short transition 
curves, by taking the ordinates starting from the P.T.G . 2 , or 
from the P.(7. 2 , proportional to the cubes of the distances, 
that at the P. G. G. being one half the offset. Or by the method 
in use with the Holbrook spirals; set up at the P. G. 2 , and 
measure the angle between the tangent at that point and the 
line to the P. T. C. 2 ; take out the deflection corresponding to the 
flat curve for the entire distance, and assume the remainder to 
be due to the transition curve; intermediate deflections to be 
made up of the flat curve deflection proportional to distance, 
and the transition curve deflection proportional to the square 
of the distance. 

Monuments should be placed at each end of each transition 
curve, so that the trackman may know where to begin his 
curvature and elevation of outer rail, and where to reach the 
full values required for the circular curve. 

120. General Case.—The transition curve adds materi¬ 
ally to the flexibility of the alignment, as already indicated. 



H _ 


Fig. 12. 



It also simplifies the field-work during location on crooked 
alignment by allowing, after the line has been fairly well fitted 
to the ground, any tangent or any curve to be studied by 
itself and shifted within the limits of the offsets if its position 
can he improved thereby. Many of the complicated field 



34 


TRANSITION CURVES. 


problems can be simplified by running trial solutions; it often 
being easy to bring the last curve within the range of offsets 
of the tangent, and then use a transition curve. To find the 
P.T. and offset F between a circle and tangent which do not 
quite intersect or touch : 

Run the curve to a convenient point A, Fig. 12, where it is 
approximately parallel to the given tangent; set up the instru¬ 
ment and measure the angle A x between the tangent proper 
and a parallel to the given tangent; also measure the perpen¬ 
dicular AB. 

Dist. to P.T. = AG = 

Offset F = AB — AG sin -X 

Z 

Or, sight from any point A near the P. T. to any point H on 
the tangent, and measure the angle z/ 2 between this line and 
the tangent proper at A; set up at H and measure the angle 
A 3 between this line and the given tangent; also measure the 
distance AH. 

A i — A 3 ± A 
AB = AH sin A 3 ; 
giving the same data as above. 

Example 1. A tangent joins a 6° curve to the left at the P.T. at sta. 
824, and a 20° curve to the right at the P.C. at sta. 831. If the tangent 
be moved 3 ft. to the right at sta. 824, and 48.5 ft. to the left at sta. 831, 
to better fit the ground, find the data required. 

Each central angle will be increased by tan— 1 = 4° 12*', length- 

<00 

ening the 6° curve by 70.1 ft., and the 20° curve by 21 ft. 

The first offset F = {R +3) cos 4° 12*' - R = 0.42. 

The second offset F' = (R r + 48.5) cos 4° 12*' — R' — 47.60. 

The length of the new tangent is best found when restaking it; it can 
be readily computed if desired. 

21. Description of Right of Way.—The ordinary 
description for right of way calls for a given width each side 
of the center line, and then describes the center line as being 
so many feet along a tangent of a certain bearing, then along 
a curve of given degree, etc. 

When a transition curve is added, it will be sufficient in 
describing the center line to note the offset thus; “ Thence 



FORMULAS. 


35 


along a tangent N. 20° W. from sta. 1081 to sta. 1086 -f 50, a 
distance of 550 feet; thence along a 6° curve to the right to 
sta. 1091, a distance of 450 feet; thence along a tangent N. 7° 
E. to sta. 1102, a distance of 1100 feet, there being an offset 
of 0.4 feet at the beginning of the curve, and one of 0.5 feet 
at the end for a transition curve to connect the circular curve 
and tangent.” The variation in transition curve for a given 
offset not being sufficient to seriously affect right-of-way lines. 


FORMULAS. 


(9), 7° =H= 2 8.65!. 


0 ° = n 2 r = ~r. 

ti 



(4) , y — IG\ G from Table II, 0° as argument. 

(6), x — l IE; E from Table II, 0° as argument. 

(5) -(10), xi, y x , F, x', y', e, e', V = x'-j- e, see Table I. 

(14)—(IT), d = —A — B, A and B from Table III, n”, n, 1° 
as arguments. 

P. G to P.G.i (circular arc) = ^100 = 

D 2 * 












36 


TRANSITION CURVES. 


COMPOUND CURVES. 


D" 

(18), l lf = h'-p. 


h = h- r - h 


V’B" ,r\ 2 
(19), /" = -- = ll- 


200 


(20), tan i — F „ jp* _ -p _ j£ 


I-L 


P.T.C.i to P.G.G. (circular arc) = --^—100. 

i — P' 

P.G.G. to P.C.i (circular arc) = ^,, --100. 


TABLES. 

Table I. 

h = length of transition curve, § 3; 

1° — central angle of transition curve from P.T.G. to 
PG. t , (9); 

= central angle of circular curve from P.G. to P.G 











TABLES. 


37 


x x ,y v — coordinates of P. G. x (3)—(6); 

x\ y r , = coordinates of point opposite P.G., (11); 

F = offset from tangent to circular curve at P.G., (11); 

V — x' -f- e — length of transition curve from P.T.G. to 
P.G .; 

e = V — x' — excess of transition curve over tangent 
from P. T. G. to P. G .; 

e' — — V — arc excess = excess of transition curve over 

circular curve from P.G. to P.G .,, § 6. 

It should be noted that for a given I linear dimensions are 
inversely as D. This will often allow one table to be used for 
interpolating in, or for extending the limits of, another. Thus 
for ^ = 50, with D = 4°, halve the dimensions for ^ = 100, with 
D = 2°, to obtain the data not given in the 4° table. 

Table II. 

<p a = n*/° = central angle of transition curve from P.T.C. 

l 

to point distant l from P.T. G., where n = —; 

y = ordinate at any point from tangent at P.T.G.) 

x = abscissa at any point measured along tangent at P. T. 01 

Table III. 

n, <p, as above for point sighted. 

ft" = instrument point = distance from P.T.G. -f- l x . 

d= deflection angle from tangent at P.T.G., in degrees, 
for all values of 1° from 0° to 60°, and for five 
values of ft". 1° is taken from Table I. 

Table IV. 

This is a table of actual deflections computed from Table 
III. The offsets F from the circular curve to the tangent at 
the P.G. or P.T. are given for easy reference, and the long 
chords, or distances from each chord point to the P. T. G. at 
the flat or zero end of the transition curve, are given to aid in 
passing obstructions and to allow of passing from one end of 
the transition curve to the other with distance and alignment 
without stopping to run the curve. 

It may be noted that the chord-point numbers are multiplied 
by the chord length to give the distances from the P.T.G ., 
and not by the total length of transition curve, as in Table III. 


38 


TRANSITION CURVES. 


The deflectionTor any chord, or the angle which it makes 
with the tangent at the P. T. G ., is given at the intersection of 
the horizontal line bearing the chord-point number of one end 
with the vertical column bearing the chord-point number of 
the other end of the chord. To find the deflections for all the 
chords radiating from a given point, run out the horizontal 
line and down the vertical column bearing the point number. 

Thus with the 50-foot-chord curve the deflections for the 
chords radiating from G. P.6 are: 6-0. 3° 00'; 6-1, 3° 35'; . . 
6-12, 20° 59'.6; the tangent at 6, 9° 00', coming from the J 
column. 

Some of the chord-point columns have been omitted, which 
requires the use of double chord lengths if the instrument is 
not at the P. T. G. The chord lengths are limited nearly to 
those allowed for circular curves for the same curvature. 

The range of the table can be extended, the same as for 
Table I, by remembering that for the same angles doubling 
distances will halve degrees of curvature, and vice versa; 
while the chord lengths will have the factor 4, because the 
doubling of distances divides the number of chords by 2, and 
vice versa. This can be seen by comparing the 100-foot chord 
curve with the 25, the 5-foot 4-degree with the 5-foot 16- 
degree, etc.; or in Table I the 2-degree with the 4-degree, etc. 


Table V. 

Radii found by dividing 5730 by D. 

Deflection angles in minutes per foot of length of curve. 

Table VI. 

Actual circular arcs for a 100-foot chord, or for 100 feet 
measured along the shorter chords. 


Table VII. 

Tangents and externals for a 1° curve with radius = 5730. 
All tables computed on the basis that P — ' -^ , requiring 


100-foot chords to be used up to 8° curve, 50-foot chords from 
8° to 16° curves, 25-foot chords from 16° to 32°, 10-foot chords 
from 32° to 80°, and 5-foot chords for sharper curves. 

In running centers for high-speed track where great preci¬ 
sion is desired the chord lengths must be changed slightly for 
the curve to accurately fit its tangents as below. 


DEFLECTIONS FOR FIELD USE 


39 


CHORD CORRECTION, OR SUM OF CHORD CORRECTIONS 

per 100-ft. Station, for 72=5730/2). Actual arc=100.007 
for the Circle and 100 for the Transition Curve. 


Correc¬ 

100-ft. 

50-ft. 

25-ft. 

10-ft. 

Tran. Cur. 

tion. 

Chords. 

Chords. 

Chords. 

Chords. 

Corr. 

+ .007 

0° 

30' 




.000 

.006 

1 



- • 


-.001 

.004 

1 

30 




.003 

+ .002 

2 





.005 

-.001 

2 

30 




.008 

.004 

3 





.011 

.008 

3 

30 

7° 

14° 

35° 

.015 

.013 

4 


8 

16 

40 

.020 

.018 

4 

30 

9 

18 

45 

.025 

.024 

5 


10 

20 

50 

.031 

.031 

5 

30 

11 

22 

55 

.038 

.038 

6 


12 

24 

60 

.045 

.046 

6 

30 

13 

26 

65 

.053 

.055 

7 


14 

28 

70 

.062 

.064 

7 

30 

15 

30 

75 

.071 

.074 

8 


16 

32 

80 

.081 


The table shows that a correction of .01 ft. for a 4° 30' curve, 
with an increase of .01 for each additional *° up to 8°, will be 
correct to about the nearest .01 for 100-ft. chords, with degree 
of curve inversely as chord length for shorter chords. 

Curves sharper than 80° should usually be run on the radius 
basis like street railroad curves. 

Table VIII. 

Circular functions to five decimal places. 

FORM FOR PRESERVING DEFLECTIONS MOST COMMONLY 

USED. 

Deflections for 2° curve; / = 1° 30'; l x = 150 ft..; F= .32ft. 
n O at Sta. 0. © at Sta. 1. 0 at Sta. .5. 

0 0° 0' 0° 30' 

.1 

.2 

.3 

.4 

.5 0 7* 0 52* 

.6 
.7 
.8 
.9 
1 


0 30 


1 30 


40 


TRANSITION CURVES. 


Deflections for 7° curve; I = 10° .50; l l = 300 ft.; ¥ — 4.57 ft. 


n 

© at Sta. 0. 

® at Sta. 1. 

® at Sta. . 

0 

.1 

.2 

0° O' 


0° 52f 

.25 

.3 

.4 

0 13 


1 32 

.5 

.6 

.7 

0 52£ 

6° 74' 

2 37£ 

.75 

.8 

.9 

1 58 


4 09£ 

1 

3 30 

10 30 

6 07£ 


Deflections for 20° curve; I = 60°; l x = 600 ft.; F — 50.33 ft. 

O at Sta. 1 


n 

0 at Sta. 0. 

© at Sta. .75. 

0 

0° 

0' 

11° 

13' 

.05 

0 

3 

12 

01 

.1 

0 

12 

12 

55£ 

.15 

0 

27 

13 

56 

.2 

0 

48 

15 

02 

.25 

1 

15 

16 

14 

.3 

1 

48 

17 

32£ 

.35 

2 

27 

18 

57 

.4 

3 

12 

20 

27 

.45 

4 

03 

22 

03 

.5 

5 

00 

23 

45 

.55 

6 

03 

25 

33 

.6 

7 

14 

27 

27 

.65 

8 

26 

29 

27 

.7 

9 

47 

31 

33 

.75 

11 

13 

33 

45 

.8 

12 

45 

36 

03 

.85 

14 

23 

38 

27 

.9 

16 

06 

40 

57 

.95 

17 

55 

43 

33 

1 

19 

m 

46 

15 


46° 15' 


60 00 


Eq. 41 ] 


SUPPLEMENT. 


40a 


SUPPLEMENT. 

22. Crossing 1 Frogs. —These may be located on tran¬ 
sition curves, especially for street-railroad work. In Fig. 15, 
let the tangent at the P. T. C. of the outer rail of a transition 
curve intersect the inner rail of a circular track at A at an 
angle EAD, and at a distance AO from the P.T.C. 



To -find the frog C. —Plat to a scale of 5 feet to an inch and 
measure AC. Compute A from AC and the degree of the 
curve. 

In the triangle ACE, EAC = EAD-\-~, and the chord AC 

is known by scale. Find AE and CE by computation, which 
accurately locates the point C of the circle with reference to 
the P.T.C. of the transition. With the distance OE as x for 
the transition find <f> and y by Table Ila. This accurately 
locates the intersection of EC with the transition; its dis¬ 
tance, y—EC, from C is the distance from the transition to 
the circle. Lay this distance off full size on EC and draw 
parallels to the sides of the frog at C to intersect, forming a 
corrective triangle. The side parallel to AC will give the 




TRANSITION CURVES. 


406 


K 22. 


distance to add or subtract for a new AC. The second solu¬ 
tion should give the desired intersection. 

The frog angle at C = BAD—A. 

Distance OC = x-\-e = l, found by Table II. 

Frog F .—Draw the radii at A and B, giving a triangle with 
the angle at A and sides R — ^g {g = gage) and R-\-hg known. 
Having B, repeat the process for finding C. 

Frog H .—Draw the normal HI to the transition; drop IJ 
perpendicular to AJ and scaje AH and KHI or <£ (<£ can be 
better found by scaling OJ and computing by Table Ila). 
Find the angle subtending the arc AH and the distances OL 
and HL, as for C. Find IK and HK from g and <£. Add 
IK to OL for x and enter Table Ila for <fi and y (if <£ differs 
appreciably from the measured vaiue, a corrected value of 
IK must be used). y+HK compared with HL will give the 
distance between the two curves as before. Drawing the 
full-sized triangle of error, as at C, will give the corrected AH 
for a second approximation. 

The difference between the l for F and that for C will give 
the distance CF between frogs. 

If both tracks are on transition curves as in Fig. 16 , find 
the x for the point C for one of the curves from the plot as 



Fig. 16. 

before and compute <£ and y from Table Ila. AB and BC 
would thus be known: compute AC and BAC. These with 
EAB allow AD and DC to be found. Adding AD to the 
distance of A from the P.T.C. of the second transition will 



§ 22 .] 


SUPPLEMENT. 


40c 


give the x for the ordinate DC. Comparing DC with the y 
from Table Ila will give the distance between the two curves 
measured along DC. Laying this off full size on DC and draw¬ 
ing parallels to the curves at C will give the triangle of error 
as before, from which the correction to x {Ax) of the first 
curve can be found for a second approximation. 

For those who prefer to work without the tables fairly 
simple relations between the coordinates x and y and the 
radius or degree of curve can be found by taking the first 
term of (3) and of (5), and eliminating k by (10). The cen¬ 
tral angle 0 can be found from either y or x by using only 
the first term of the equation under (3) or under (5). With 
a slide-rule results are readily obtained and the approximate 
equations are accurate enough for many purposes. 

For complicated street-railroad layouts a connecting curve 
made up of short circular arcs compounded is more common 
as it lends itself rather more readily to computation, but no 
difficulty need .be experienced with the true transition curve 
along the lines as indicated above. 

Example 1. Given Fig. 15, AO = 112 ft.; angle EAD = 47° 01'; 4° 
circular curve on center line; 200-ft. transition to a 20° curve on outer 
rail; to find the crossing frogs. 

Frog C.—AC = 6.2 ft. from a plot to a scale of 5 ft. to an inch. 

A for + C = 4°X .062 = 0°.248 = 0° 15'. .\ BAC = 47° 8*'. 

CE = 6.2 sin 47° 8*' = 4.545; AE = 6.2 cos 47° 8*' = 4.217. 

x= 112 + 4.217 = 116.217. 7by(9) = 20°. 

Table Ila, with x = 116.217, 7 = 20, gives 0 = 6°.771. y = 4.580. 

y — CE = 4.580 - 4.545 = .035. 

By laying .035 off on EC and completing the corrective triangle it is 
found that AC should be increased .055. 

AC = 6.255 gives J= 0° 15'; BAC = 47° 81'. 

CE = 4.585; AE = 4.255; a = 116.255. 

This value of x gives <£ = 6°.776 = 6° 46V'; y = 4.585, =CE. 

.*. Frog angle = 47° 01'-(6° 46*') + (0° 15') = 40° 29*'. 

Frog distance from P. T. <7. =x + e= 116.255 + . 163 = 116.42 ft. 

Frog 77 .—AH by scale=13.6. 

a; by scale = 121.8, giving <£ = 7°.45 = 7° 27'. 

A for AH = 4X. 136 = .544 = 0° 32*'. £.477 = 47° 17'. 

HL = 13.6 sin 47° 17'= 9.992; +7,= 13.6 cos 47° 17'= 9.226. 

77C = 4.71 sin 7° 27'= .611; 777£ = 4.71 cos 7° 27'= 4.670. 

a; = 112 + 9.226 + .611 = 121.837, gives 0 = 7°.447; y = h. 280. 

2/+ 777: = 9.950. 9.950 -HL= -.042. 

.042 laid off on HL will decrease AH by scale .064. 

AH = 13.536 gives A= 0° 32*'; £+77 = 47° 17'. 


m 


TRANSITION CURVES. 


[§ 22 . 


RX = 9.945; AL = 9.182; IK and HK as above. 
x= 121.793, giving cf> = 7°.442-7° 26V; 2/ = 5.275. 

V + HK = 9.945, = IIL and checking the frog-point. 

Frog angle = 47° 01'-(7° 26£') + 0 o 32^'= 40° 07'. 

Distance between frogs, (7#= 13.536-6.255 = 7.28 ft. 

Similarly the frogs at F and G can be found and the remaining dis¬ 
tances. The distance between H and G can best be found by coordi¬ 
nates, having the distance of each from the tangent OJ and the projec¬ 
tion of each upon this tangent. 


Table la. 


This is tabulated for radius, rather than for degree of circu¬ 
lar curve, for convenience in running sharp street-railroad 
curves. The remarks for Table I, pp. 36, 37, apply, except 
that in comparing the length of the circular curve from P.C. 
to P.C.! with that of the corresponding portion of the transi¬ 
tion curve the actual arcs (I°R arc 1°, and ^-l') are used, 


giving 



to P.C. V 


= excess of transition over circular curve from P.C. 

§ 6 . 


Table Ila. 


This table is for coordinates when the abscissa x, rather 
than the distance l along the curve, is given. 

<j> = central angle of the transition curve between the P. T.C. 
and the point with coordinates x and y, I being its largest 
value at the P.C. t . 

l \ = length of transition curve for central angle /. 

From (8), = 


Putting in the value of l from (6), 


<£ = 


lx 2 x 2 

h 2 (1 


L 1 


where 


1 +F- 


(1 -E) 2 ' 


From (4), y = 1C, - j~x - Gx. 

F and G vary with as seen from the table. 





TABLE I. 


- ' 

D O O O 
TOOOO 

T"H 

O O O O O 
WTfCOGOO 

l—l T—i rH r—1 03 

O O O O O 
W^COODO 
03 03 03 03 CO 

o o o o o 

03 TP CO 00 O 
CO CO CO CO Tp 

ooooo 

03 ^ ID 00 O 
Tp Tp Tp tp aO 

OOOOO 
03 TP CD GO O 
aO aO aO aO CD 

OOOOO 

O* TP CD GO O 

O CD CD CD L- 



03 tP l> rH 
O rH 

CD-HOOiOTf 
rH o* 03 CO Tp 

CO CO Tp CD 00 
AD CD L- 00 05 

03 CD rH i>- Tp 
r-i C3 iO N 

03 rH rH rH 03 
05 rH CO AO t'* 

tP 00 03 CD 03 
05 r -1 Tp tO 05 

05 CD Tp CO CO 
t— t tJ* t— O CO 






rH rH rH rH rH 

l-H 03 03 03 03 

03 CO CO CO CO 

Tp TT TP AD AO 






05 05 05 

05 05 05 05 05 

00 00 GO GO t> 

J> CD CD AO 



OOOO 
03 CO tP lO 

O O O O O 
Oi-QOQO 
1—1 

o o o o o 

rH C3 CO TT aO 

T —1 rH r-i r—i rH 

^ 05 05 05 
CO i."— i — GO 05 

l-H r-l r—i r rH 

05 05 05 05 05 
O T—1 03 CO tP 
03 03 03 03 03 

05 C5 05 05 05 
AD CD A- GO 05 
03 03 03 03 03 

05 05 05 05 05 

O rH 03 CO TP 

CO CO CO CO CO 



CO GO Tji 03 
O th 03 

WWOhN 
CO Ttfi lO GO 

OOCOhcO 
O 03 tp i> 05 

CO 03 CO AO 05 
03 AO GO r-( 

AO 03 th 03 AO 
GO 03 CD O TP 

05 iO CO CO rf 
GO CO 00 00 GO 

i- 03 00 i- t- 
CO 05 TP O CD 


fc* 


i 

iH rH rH rH rH 

03 03 03 CO CO 

CO Tp TP AO AO 

ID CD CD i> t- 

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CO GO 1-H CO CO 
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53 






















































































ri 

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CO ^ o 

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CO C- 00 05 O 

t— t 

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CO ri rF CO GO 
O r^ 07 TF CO 

00 co 07 CO AO 
05 CO t-r r-i CO 

05 N GO CO r-i 
t-h N CO O N 

rH 

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07 CO CO ''F O 

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CO CO FF AO AO 

cd 



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05 

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05 00 *> CO FH 

05 05 Tf 05 

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05 

AO O AO O t* 
r -1 t—i 07 07 

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05 fF C5 fF 05 
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fF GO CO GO 07 

ao ao co co i> 

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r: 

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05 05 00 AO r -i 
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05 CO aO ^F fF 

CO 00 05 o o 
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GO AO 07 05 AO 

00 AO CO ri T—I 

07 GO CO 00 rn 
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CO 

05 

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rH 

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54 





























































































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GO iO CO 
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if to O AO CO 
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02 CO in CS If 
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1-00 
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02 02 02 




O 05 05 
tO CS rf 

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CS rf 05 rf GO 

C5^NO 
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CON 
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to O to 02 O 

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go io i— 

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t- If 00 
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csx.cs 

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rf 05 rf 

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55 






























































































TABLE I 


P 

o 

O 

Oi 


T-\ 

O AO O AO 
r-« t— 02 02 

OAOOAOO 
W CO TT Tt o 

AO O AO O aO 
AO CD CD L- i- 

O aO 
GO GO 


O AO O AO 

i »—i <M <M 

O AO O AO o 

02 CO Tf ^ AO 

AO O aO O AO 
AO CD CD L— i.'- 

O AO 

GO GO 


^5 

i’ONN 
O r-. — (M 

05 CO 05 f- l> 
CO AO CD 00 o 

C 5 G'? i- CO 

(M aO L- O CO 

CO Tf 
CD 05 


»0 rH O t—( 
OmWCO 

^ O 00 GO O 
Tf CD L- 05 (M 

-V O GO GC O 
Tf t- 05 O? CD 

CO OO 

05 02 



T —1 

ruH H (M Ol 

09 oi 



1 —H 

t-h t-h t—i O? O? 

02 CO 



O 05 GO t- 
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iO <M co co i> 

05 T* GO CO l- 

O —• O CO 

Of CD o co 

GO 05 

O CO 


O 05 GO CD 
O Tf 05 TT 

CO 05 -rf 05 t-h 
05 CO 00 OJ t- 

r- 05 CD rH CO 
r- Tf X CMO 

CO O 

00 r-H 


H 

AO 05 02 

T“H 

Tt' o- 05 o>.rr 

*—• < t-i CQ O* 

O 05 M Th CD 
O^ <M CO CO CO 

05 — 
CO Tf 


AO J> 05 <M 

r —1 

Tt 1 i.'- D Ol rp 

th rH n Ol W 

05 — rf CD 
O? 02 CO CO CO 

00 T-H 

CO ^ 


fc. 

C 5 o AO Tfi 
O (M CO AO 

GO CD 05 AO CO 
J> O CO i- 

TH o O* C 5 o 
CD t—< CD r—• GO 

CO CO 
rr i—» 


O O ? 05 O? 
r~< (M CO CD 

05 O CD 00 Tf 1 
00 M AO 05 Tf 

TF O 05 CO rH 
05 AO O L- ^ 

rr O 
rn 05 




lH 7 —( r—1 

(M CO CO Tf ^ 

AO CD 



rHrHHOI 

OlCOrj-rf aO 

CD CD 

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w 

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03 

D 


T-J C* 

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t 

CO CO 05 CO GO 

T—• T—< 

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CO —' O O y-* 
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# o • • • 

Tf 05 
L'- GO 

• • 

r-« (M 
O 
• 

Tf J> rH CD O? 
t-h rH O-? 

r-* O O CO 05 
CO TT AO CD L- 

CD CD 

05 «-H 

O 

o 

O 

02 


rH 

O 

• 

Hr-QICO't 

AO 00 t—* CO 

T— 1 t—« 

• • 

CD 05 

T—* T-H 
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O 

o 

CD 

CO 

T-H 

tH 

o 

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r-< O? CO Tf CD 

L- 05 {> 

T-H T-H r-H 
• • * 

rH 

05 

t—• O? 

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rH 

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AO 00 C 5 i> 
CO l- CO »— 

t-H 02 

0 ^ Tf O? CD CD 
y~* OX AO 05 *0 

CO Tj* AC CD GO 

rH (M N CD D 
CO Ot TT 

O M Tf CD 00 

1 —< T—' T—' T—< TH 

AO co 

'M GO 

— CO 

(M (M 


O 05 GO CD 
rf X aO tt 

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rH t-H t-H t-H 

04 rtf !>. 05 rH 
04 04 04 04 CO 

GO 

to 

II 

fe. 

05 to rtf rtf 

04 rtf CO 00 t-h 

rH 

to 00 to co o 

Tf N H to O 

rH t-H 04 04 CO 

CO 04 to 04 CO 
qCOONOCO 

10 ’ * ‘hh 

II 

NNO^OCO 
CO O tO 05 rtf 

rH 04 04 04 CO 

CO CO 04 rH rtf 
. CO tO 00 rH rtf 

rtf t-H rH 

II 

rH CO O rH CO 
OC 04 04 

rH 04 04 CO CO 

Radius 

\) 

t-h 04 04 co to 
o 
• 

t-H rH 04 

O 

• 

05 co co r— r— 

rH t-H 04 04 
• • • • 

04 CO rtf to CO 

Radius 

.01 

2 

.01 3 

1 6 

2 9 

COf^ 04 OOt^ 
rH rH 04 04 CO 

CO ^ tO CO GO 

m 

P rH 04 rtf CO O 

W O rH 

Q • 

<1 

HH rH rH 04 04 

O 
• 

tO t—h to co 

T— 04 04 CO rtf 

CO rtf CO O 

rH 

• 



UOONOO 
t—11 ''. iO »0 tO 

CO 05 to to 
NOtOnoO 

co oo to 

CO O 05 O 04 

10 05NOC0 
CO T-H 00 CO CO 

rtf »0 rtf 05 04 
rtf 04 04 CO t'r 

HNOOtON 

(MOOOON 


535 

HH04COTJH 

to 00 O H 

rH rH 

t-h 04 O'J rtf tO 

CO GO 05 r-H CO 
rH rH 

t-h 04 CO rtf tO 

00 O 04 rtf 

rH rH rH 



'tOOOO^ 
05 GO CO 1>- tO 

CO GO 00 t-H 05 
C0ON^05 

04 rtf CO !>- CO 
05 00 to CO 

05 CO CO CO rH 

O N co co co 

HC4 00010 
05 00 CO to 04 

CO rtf O CO 

05 to O to GO 


H 

05 rtf C5 rtf 05 
rH 04 04 CO CO 

rtf 05 CO CO 04 

rtf rtf to tO O 

05 rtf 05 rtf 05 
rH 04 04 co CO 

^ oo co r- 04 
rtf rtf LO to CO 

05 rtf 05 rtf 05 
t-h 04 04 CO CO 

CO 00 CO rH 

rtf rtf to .to co 


o 

GO tO 04 C5 CO 
00 CO GO 03 t— 

COON^C 

04NHC0H 

CO 04 05 «0 04 
rtf CO rH O 05 

GOtO T-H Hfl 

co to co 04 

ONOOOH 
rtf LO CO GO 05 

CO rtf to 00 

O rH 04 CO rtf 


►•N 

05C4rfHN05 

rH t-H t-H t-H 

04 ^ 05 04 

04 04 04 04 CO 

HTfiNOC) 
rH rH rH 04 04 

lOOOHTfN 

04 04 co CO CO 

04 *0 CO t-h —tf 
rH t-H r-H OJ 04 

COHrtNO 

04 CO CO CO rtf 


V. 

CO t-H 0> rH rtf 
thC4CO^O 

OOCO O 05 05 
CO GO O rH CO 

iOCOtHNH 
t-H 04 CO rtf CO 

N IO to GO 

|>. 05 T-H CO to 

^ 05 CO 05 

rH 04 CO to CO 

1>- CO I>- rH CO 
00O04L0N 




rH rH t-H 


rH rH rH 


rH rH t-H tH 


V 

05 05t'- CO CO 
05 rtf 05 rtf 05 

OOC4NH 

^ oo co r- 04 

05 00 CO r^ H 
05 rtf 05 rtf 05 

t''- 04 CO 05 04 
CO 00 04 co rH 

05N000 05 
05 rtf 05 rtf CO 

CO I>- 05 rH rH 

CONhOO 



05 04 rtf 05 

t-H t-H t-H t-H 

04 rtf 05 04 

04 04 04 04 co 

05 04 rtf r^- 05 

r-H rH t-H H 

04 rtf 05 04 

04 04 04 04 co 

05 04 rtf 05 

T—H t-H r-H rH 

04 rtf 05 04 

04 04 04 04 CO 


fc, 

NNOC4N 
04 rtf CO 00 O 

tONWOO 
CO co O rtf 00 

tH CO 05 rtf CO 
CO rtf CO 05 04 

to rH rH tO 04 

to 05 CO 04 

tO rtf GO CO GO 
WiONOCO 
• • ♦ • • 

rtf tO O 05 rH 

NhCOOO 

• • • • t 



• • • • • 

t-H 

T-H t-H 04 04 04 

t-H 

T—1 rH 04 04 co 

t-H t—H 

rH 04 04 CO CO 


\> 

t-H t-H 04 CO CO 

o 

• 

GO 04 tO 05 rtf 
t-H rH rH 04 
• • • • 

rH 04 CO tO I>- 

o 

• 

r-H to O co H 
rH rH 04 03 CO 

rH 04 CO CO 05 
O 
• 

rtf 050 04 O 
t-h h 04 CO rtf 

• • - • i 

04 

CO 


t-H t-H t-H 

o 

• 

04 04 CO rtf tO 

• 

^ r-H t—1 04 

tO o 

II 

04 CO rtf tO N 

• 

GO rH t—H rH 04 

rtf O 

II 

CO rtf to I>- 05 

II 




rf) 


CO- 


CO 

p 

M 

Q 


r-1''— r-H r^- r- 
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05 rtf 04 h CO 

CO CO O to rH 

P 

h CO 04 O to 05 

P 04 C5i^t^ 00 

NCNNO 

hcOhXN 

p 

HH OO CO O ^H 05 

P CO H T—1 04 rtf 

04 rH CO CO O 
05 to 04 rH 04 

Ph 

53) 

t-H rH 04 CO rtf 

to CO OO 05 rH 
rH 

P H h 04 W Tin 

CO t'- 05 C 04 

TH t-H 

pj h 04 CO rtf to 

CO 00 O 04 rtf 
rH rH rH 



to O 04 04 05 

05 05 co r— to 

rH 05 CO rH rtf 
rtf t-h 05 CO 04 

CO co co CO to 
05 OO CC rtf 

CO rf 05 05 

04 05 tO t-h CO 

T-H CO r-H rtf rH 

05 GO t"- to co 

04 CO 04 O 00 
OC0 04NO 



05 rtf 05 rtf 05 
rH 04 04 CO CO 

rtf 05 CO CO CO 
rtfi rtf to to CO 

05 rtf 05 rtf 05 
rH 04 04 co CO 

rtf 00 CO GO 04 
Tf 1 Tji LO tO CO 

05 rtf 05 rtf 05 
rH 04 04 CO CO 

rtf CO CO t>- 04 
rtf Ttf to to CO 


o 

Tf iO cO N CO 
04 to GO rH -rtf 

05 O rH 04 CO 
N rH rtf £-— O 

rH CO 04 t"— 04 
CO 04 05 to 04 

CO OOCO 00 
00 to T-H GO rtf 

rtf 04 O 05 

05 05 05 00 GO 

CO ^ CO H 05 
00 00 00 00 


^-h 

05 T-H CO co 00 

t-H t—h t-H t-H 

OCOiONO 
04 04 04 04 CO 

O CO tO 00 rH 
t-H t-H r-H t-H 04 

COCOOrHrf 
04 04 04 CO CO 

HrtfNOCO 
rH t—H t-H 04 04 

CO 05 04 to OO 
04 04 CO CO CO 



o to o to O 
04 04 CO CO rtf 

to O to O to 
rtf to to co co 

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04 04 CO CO Hf 

to o to o to 
rtf lO to co CO 

O to O to O 

04 04 CO CO rtf 

to o to o to 

Ttf to to CO CO 


> 


56c 




































































































TABLE I a. 



»H 

*0 OlOOO 
HCqiN CO CO 

0*0 0*0 O 
Tf< lo *0 cO 

LO 0*0 0*0 
T-H CM CM CO CO 

0*0 0*0 O 
Tf *0 *0 CO 

*0 0*0 0*0 
T-H CM CM CO CO 

0*00*00 
■Tf LO *0 CO 


V 

«3> 

T-H o O Tt< O 
T-H CM CO CO 

OOOOOrfH 
L- 05 CM ' r + i L"— 

CM CM ^ 05 CO 
t-h CM CO CO 

OC0CM05N 

00 o co *o oo 

COt^NCOh 

HCMCO*ON 

CM *0 T-H o O 
05 T-H Hf {>. o 



t-H t-H t-H 

. 

t-H t-H t-H rH 


rH T-H T —1 CM 


H 

05 CO N- o 
^ 05 ^ C55 ^ 

*C 05 t-H CM O 

OO CM N- t— i iO 

05 co*o co oo 
05 HtH 05 CO 

CM *0 *0 CO 05 
OO CM CO O CO 

CO 00*0 rH *0 
^ 05 ^ 05 CO 

OOOO^N 
CM O 05 CM 


05 CM tJH 

T*H T—I rH 

05 CM ^ 05 

T—1 CM CM CM CM 

N 05 Cl HjH N 

T-H T-H T-h 

05 CM ^ 05 

t-h CM CM CM CM 

i>- 05 CM N- 

t-H r-H t-h 

05 CM ^ O 05 
t-h CM CM CM CM 


&, 

CM O H 00-t-h 
CM ^ CO CO CM 

00 O CO CO T—< 
*0 O 05*0 

h^h GO 00 CO 
Cl r}H O 05 CO 

CO 05 o *o O 
N hi N Cl GO 

NN^COt^ 

CM-^NO^ 

GO 00 CO CO N- 

00 CO 05 *0 t-h 

42. 


t—H 

t-h CM CM CM CO 

rH 

rH cm CM CO CO 

T-H rH 

rH CM CM CO 




• 




Radius = 

V. 

<s> 

tH cm CM *0 CO 
o 

• » 

HHCM 

o 

CM^ rf rH CM 
t-h t-h CM CO ^ 

CO -rf *o 00 

Radius = 38. 

.01 

2 

.01 4 

1 6 

2 .10 

*0 005 0 0 
T-H CM CM CO *0 

CO *0 O 00 T-H 
t-H 

*o 

°? CM T-H CM 

II O H 

p 

P 

P T-H t-h cm CO 

3 ° 

Ph 

co *o CO O t-h 
t-h CM CO HjH O 

rfiONOCM 

T-H H 

• • 


T-H 

05 00 CO TfH O 

00 *0 rjn 1 C co 

*0 Tf N 

CM 00 O CO N- 

00*0 CM O 05 

05 r-^ 05 cm 

ooco CO 00 o 

00 O co N H 

N- 05 *o CO CO 
O CO 05 CM N- 

Tt O CO GO o 
CO t>- CM 



H(MWrf 

co b- 05 T— H co 

t-h r-H 

r-H CM CO *0 

CO CC O CM *0 

T-H 1-H T-H 

r-H t-H CM H" 1 *0 

NOHCOO 

t-h r-H t-H 


T-* 

*0 05 CO CM 05 
05 00 N- CO CO 

OC0C0 05H 

t-h cm co o 

^ co co ^ 

05 OO b- *0 CM 

hlO OOON 
05 ^ 00 T-H CO 

CO '•& oo *o CO 
05 OC CO Tt< T-H 

t-h GO CM T-H 

Nh»ONN 


H 

^ 05 ^ 05 ^ 
t-h T-1 CM CM CO 

OicooocMr- 
CO Tt< Tji LO *0 

05 Hf C5 

T-H l-H CM CM CO 

oo co N- CM co 

Cr J Hf *0 *0 

t-h t-h CM CM CO 

00 CO N- t-h lO 
CO^rfiOtO 


0 

co^iocor- 
CM co O Tji 00 

00 05 O T-H CM 

CM w T— LQ C5 

T-H 00*0 CM 05 

CO O CO CO 

ociocco 

r-H 05 CO CM 

GO N- CO *0 *Ox 
CM CO xf *0 CO 

^ CO CM CM H 
NOOOJOh 


s 

O CO l>- O CO 
r-iHrH CM CM 

o ^ r- o 

CM co CO CO Hfi 

T-H *0 00 CM co 

T-H rH T-H CM CM 

OCONhiO 
CO CC CO "'t 1 

CM O O ^ 00 
T— T-H CM CM cm 

CM CO 0*0 05 

COCO^^rjH 


V 

iH c; 05 CM N 
rHrHCMTf lO 

^■^10 05^ 
£'— 05 t-h CO CO 

t-h t-h t-H 

CM T-H CM CO CO 

T-H CM co ^ O 

CM co co T-H Oi 
COOC'lON 

rH t-H rH i-H 

CO CO CO t-h 05 
r-r CM CO *0 CO 

O CO 00 O O 

05 t-h CO CO 05 

rH rH t-H t-H 


V 

H 

005 CO ^ T-H 
LO 05 ^ 05 rft 

COrH^lOTtl 
00 CO T-H IO 

05 00 CO CO 05 
rp 05 Tp 05 CO 

CO CO OO 00*0 
OO CM CO O ^ 

OO 00 *0 T-H CO 
^ 05 TjH C5 CO 

05 t-h O CO CM 

CM CO 05 CO 


1>- 05 CM ' r ±< L- 

l-H t-H rH 

05 CM ^ 05 

T-H CM CM CM CM 

t>- 05 CM N- 

T-H T—H r-H 

05 CM Th N- 0-> 
t-h CM CM CM CM 

N- 05 CM ^ N- 

tH rH rH 

05 CM CO 05 

t-h CM CM CM CM 


&, 

H O0 05 to *0 
CM CO *0 00 rH 

OO^CMiO 
*0 05 CO 00 co 

CO CM *o CO N» 

CM CO 05 CM 

COOONOCO 
CO O *0 t— CO 

CO CO CM CO tH 

CM Tt* IN- o 

^ CM *0 CON 

GO CO GO ^ O 



rH 

rH t-h CM CM CO 

rH 

T-i CM CM CO CO 

t-H t-H 

rH cm CM CO Hf 


\> 

rHCO^ON 

o 

• 

HIOHQC5 
t-h t-h cm Cm co 

T-H CM CO CO 05 

o 

• 

TfOcO^iO 
t-h CM CM CO Hfi 

dd^NH 

O t-H 

• • 

^ CO CO CO 
rH CM CO ^ *0 

• 

ii 


t-H t-H CM 

o 

• 

COr^*OON 

5 T-H t-H CM 

II 

CO^fcOOOO CO rH CM co 

H co o 

• II 

^*ON 05CM 

rH 

• 

II 








to 

P 

M 

r~ 

lOrHlO00O5 

co *o co co *o 

CO *0 'Tf 05 
05 lo CM t-h t-h 

CO 

p CO CO 00 tH 
^05O*0NO 

*0 *0 CO 05 t-h 
*0 CM T-H t-h Hf 

P 

h t^^NhO 0 
^Q00 00rH*O 

*0 CM GO CO CO 

CM T— T-H 00 

P 

<1 

Ph 

s 

rH CM CO XjH 

*0 1>- 05 t— i CO 
t-h t-h 

^ T-H CM CO *0 

Ph 

CO 00 O CM Tf ^HHCM^IO 

N* 05 th CO *C 

rH t-H rH 



COOOiCiO 

05 05 CO CO ^ 

OO-^tHON 
t-h CO GO CM 

*ONCOOO^ 

05 OO N- *0 CO 

T-H O 00 CO T-H 

OOO^N 

CO *c o oo oo 

05 GO t>- rH 

00 N- GO CO 

N- CM CO 00 05 



TjH 05 -rfH 05 HH 
rH t-H CM CM CO 

05 CO 00 CM N- 
CO Tt 

^O5^05rt< 
th t-h cm cm co 

05 CO 00 CM CO 
CO rji *o *o 

tJh 05 tJH 05 
t-h t-h CM CM CO 

COCONhiO 

CO ^ LO *o 


o 

N- CM 00 CO 05 

N O CM >0 N 

05 COCO 05 CM 
t-H t-H rH CM 

Tt^OlCHN 

ocooooo 

CO 05 CM *0 05 
CM CM CO CO CO 

t}<CM005N 

NCOO-^O 

OTjiNrHlO 
t-h 1-H rH CM CM 

*OCOHC5N 
CO CM 00 CO 05 

00 CM *0 05 CM 
CM CO CO CO ^ 

■^CMONiO 

05 05 05 00 00 

HIC05CON 
rH rH rH CM CM 

COhG5N*C 

00 00 N- 

T-H LO 05 CO N 

CO CO CO Tf 1 rf 


rH 

lOOlOOO 
r-H CM CM CO CO 

0*0 0*0 o 

TfH LO *0 CO 

*o 0*0 o*o 

T-H CM CM CO CO 

0*00*00 

TtH *0 *0 CO 

• 

% 

*0 0*0 0*0 

T-H cm CM co CO 

0*0 0*00 

Tf LO *0 CO 


56c? 



















































































































l»s 

6s 

+ 

CS l=s 

►—i 

II 

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c 6 
H 
Eh 

£ 


fi 

PH 

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l-H 

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t-H 

CD ^ CM CM 

CO^iOON 
CD CO O CO CO 

O o O O CO 
GO 05 O t-h t—i 
<©ONNN 

CO 000 CO CO 
CM CO ^ LO co 
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OO CO CO co CO 

NOOOOh 
B- 00 00 

CO CO co go O 
CM CO ^ LO 

OO GO 00 00 00 

OOOO 

OC' 05 O T-H 

00 OU 05 05 

6s 

CO ^ CO CO ^ 
r ^ 03 t'— 

co co ^ ^ ^ 

H t-H t—H r—( T-H 

O 0*0 *o O 
t-h io 05 ^ O 

rH N rH LO 

*0 *0 *0 CD CO 

T-H r-H t-h T-h T-h 

05 CO CM *0 CO 

*0CM05C0tJ1 

00 CM *0 C5 CO 

cor^r^t^oo 

t-H t-H t-H t-H t—1 

CO LO 00 CO 05 

CM t-h O O O 
NrHlOOCO 

00 05 05 05 O 

r —1 T-H T-h T— CM 

N- O 00 CM t-h 
^hCOtTNO 
NrHlOO^t 

O T-H T-H t-H CM 

CM CM CM CM CM 

CD CO T-H rH CO 

co i>- cm r— cm 

OOCMNthO 

CM CO CO ^ 

CM CM CM CM CM 

1 

o 


O 


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^ CO CO CO GO 

t-H t-H t—H t-H t-H 

CO CO CO CO CO 

OO GO o O O 
t-h rn CM CM CM 
CD CD CO CO CO 

O^T^TfCO 
CM CM CM CM CM 
co co CO CO CO 

CO CO 00 00 0 
CM CM CM CM CO 
CO CO CO CO *0 

O CM CM CM ^ 
CO CO CO CO CO 
CO CO CO CO CO 

CO co OO 

COCO CO CO 

co CO CO CO 

• 

0) 

cm Ob n. *o co 

IOIOCONX 

05 CM *0 00 T-H 
CONNI^OO 

CM CM CM CM CM 

CM t-h O O O 

05 O t-h CM CO 

OO GO 05 05 05 

CM CM CM CM CM 

O O CM ^ CO 
*0 CO N OO 

O CO CO 05 CM 

O O O O H 

CO CO CO CO CO 

05 CM *0 05 CO 

05 T—t CM CO 10 
*0 05 CM *0 GO 
t-h t-h CM CM CM 

CO CO CO CO CO 

00 CO 05 LO T-H 

CO OO 05 t-h CO 

t-H 1^. r-H Tf4 

co co co ^ ^ 
co co co co co 

GO *0 CO t-h O 
^ co or 0 CM 
NOCONO 

*0 *0 *C co 

CO CO CO CO CO 


o 


0 


0 


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*0*0*0 

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co 

^ Tt« 

48 

49' 
50' 

T-H CM 

*0 *0 

co ^ *0 
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co b- 
*0 *0 

OO 05 0 

LO *0 CO 


o 

OOOCsjOOO 

CO ^ ^ ^ 

TfOOOOTt 
CO tt 1 *0 cO 

Tt* rt< Tt* 

CM O O 00 

N- 00 05 05 © 
Tf XT' ^ T-js LO 

CM co CO ^ ^ 
t-h r—! CM CO 

10 *0 *0 *0 *0 

CM O CO CO rt 

IOCOONCO 

10 *0 *0 *0 *0 

CM CM CM 0 GO 

05 O t— 1 CM CM 
lO O CO CO CO 

• 

NCOO^O^O 

CO CO CO ^ lO 

CO O CM ^ GO 
lO CO O O CO 

o 

05 CO CO 0 00 
co 00 0 CO LO 
COOCO*ON 

CO l> L - 13- I>- 

O CO cO t— O 

05 CM co t-h cO 

05 CM B- 05 

r - GO 00 00 00 

0 

cqoo^ON 

T-H 0 CM 05 *0 

CM ^ B- 05 CM 

05 05 05 05 O 

t-H 

C5 lO LO 05 1>- 

CMOfOCOiO 
lo 00 0 CO co 

O T—H T-h T-h 
r-H t-H t-H t-H t-H 

05 lo CM CM 

Tf tT iO CO 

05 CM *0 CC t-h 
t-h CM CM CM CO 

t-H t-h t-H r-H t-H 


o 


0 


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CO CO OO GO GO 
OOOOQ 
LO lO 1C »0 

OO O O CM O 
05 O O O O 
*0 O CO CO CO 

CM CM ^Tf rf 
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O cO cO co co 

-rf ^ CO CO 

OOOOO 
cO O CO cO cO 

0000 0 000 

O O T-H O T-H 

co CD co cO co 

O CM CM CM CM 

t—H t-H t-H t*H ^H 

co cO CO CO CO 

• 

Cb 

OO co ^ CO (M 

GO O0 OO OO O0 
OOrHTfNO 

00 00 Q0 05 

t-H t—h t-H t-H t-H 

—h O 03 O t-h 

00 00 00 00 00 

CO CO 05 CM *0 

05 05 05 O O 
t-h t-h t-h CM CM 

t-h cm co * 0 N» 

00 00 00 00 00 

OOtht^NO 

O t-H t-H t-H CM 

CM CM CM CM CM 

05 T-h co *o 00 

00 05 05 05 05 

coco 05 CM *0 

CM CM CM CO CO 

CM CM CM CM CM 

H LO 05 ^ 00 

O O O H ^H 

05 CM *0 00 T-H 

CO Tf ^ ^ *0 

CM CM CM CM CM 

CO GO ^ O CO 

CM CM CO Tt* ^ 
rt^NOCO CD 

LO *0 CO 0 0 

CM CM CM CM CM 


o 


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t-H CM 

CO CO 

CO ^ *0 

CO CO CO 

CO B- 
CO CO 

00 05 0 

co co ^ 

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43 

44 

45 

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cocqoo^o 

05 O O t-h CM 
t-H CM CM CXI CM 

CO CM O 00 

CM CO 'B lO 

CM CM CM CM CM 

O CO CM 00 CO 
COONNOO 
CM CM CM CM CM 

t+OO^CM 
05 O O t-h CM 

CM CO CO CO CO 

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CM CO ^ ^ *0 

co co co CO co 

CM O CO rJH O 
co 00 05 

CO co co co co 

F .1 

*0 CO -T^ 00 *o 
r— n- oo 

r^lOONOO 

t-H t-H t-H t-H t-H 

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05 O CM ^ 0 

O t-h CM CO "Tt* 
t-h CM CM CM CM 

LO *C GO ^ CO 
c - CM *0 05 CO 
LO B- 00 05 T-H 

CM CM CM CM CO 

O 

co co co co co 

CM CM GO 

cm ^ *0 b- 00 

CO CO co co CO 

00*0 *0 05 

•^t 1 O l>- T-H 

OCMCO*ON 
^ ^ ^ ^ 

O 

CO CM O CM 

05 co lo ^ 

OOOCM CO 

Ttuo *0 *0 *0 


o 


O 


O 


O 

OOcOOcO 
OO GO GO CO GO 

lo *o »o *o *o 

co 00 00 000 
00 cn or <y 05 

10 *0 *0 *0 *0 

00 GO OOO O 
GO OO 00 05 05 

10 *0 *0 *0 *0 

0 0 0 CM CM 
05 05 05 05 05 

*0 *0 *0 LO *0 

CM CM CM CM ^ 

05 05 05 05 05 
LO *0 *0 *C *0 

^t 1 -rf ^ CO cO 

05 05 05 05 05 

LO *0 *0 *C *0 

y-H 

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CO 05 CM *0 oo 
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05 05 05 05 O 

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t-h 00 CM co 

OOCOC^N 
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(***~^ <f~b t-h t-H t-H 

T-h t-H t-h t-h t-H 

T—1 LO 05 CO CO 
B- 0*0 *0 

05 CM *0 CO t-h 

T-H CM CM CM CO 

t-H t-H t—H t-H t-H 

CO GO CO 00 Tt* 

^ CO CO CM CM 

N- O CO cO 

CO CO ^ ^ 

t-H t-H t-H t-H t-H 

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CM T-H T-H O O 

05 CM *0 GO t-h 
rf *0 *0 *0 CO 

t-H t-H t-H t-H t-H 

T-H 00 *0 CM O 

O 05 05 05 05 

^ co 05 CM *0 
COCOCONN 

t-h r-H t-h t-H T-H 


• • 

o 


O 


O 



10*0*0 

LO *0 

lO *C 

*0 *0 

LO *C *C 

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27 

28 

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t-h CM CM co 

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co co ^ *0 *0 

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CO CO N 00 05 

t-h t-h t-h t-h t-H 

l-H 

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57 












































TABLE III.—DEFLECTION ANGLES d = -A - B. (See also Table IV.) 



58 






































































































































* . 
















TABLE IV.—DEFLECTIONS. Zero on first tangent. 

100-foot Chords per 1°. 




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TABLE IV.—DEFLECTIONS. Zero on first tangent. 
25-foot Chords per 1° .—Continued. 



W^CHOJ 



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H' if. GO W 

o o cd 05 

CO rH ic O 


rH 

o 

SD 00 0? CO CD 
rH CO aO r-t CO 


O 

o 

rH 

rH ©S <M CO CO 

rH AC 1 C CO 

GO 05 o rH o* 

*—’ rH rn 

CO rH AD i> 

T— ■ rH t— > r— t 


rH 

rH r-H rH O? O? 


t-h 

28'.5 

42 

58.5 

AC 1C 

0000^0 
r-t rH O CO O 

AC rr 05 CO 

O GO GO r-t cd 
rH r-t ic rH CM 

i- r ID D 

• • • • 

i-OiOCOr 
rir OOO 

O O t- AC 
• • • 

05 SD 1C GO 
Oth^CO 


05 00 05 
rH OV? CO 

WN^WTf 

ic O CJ rr O 


o 


o 

rH rH CM CM CO 

CO rH rH AC CD 

i- GO 05 O t-h 

'T—H 

C? CO rH iC 



rn rH rH O'? 


o 

• 

eg 

aO lO 

SO CO rH t - 
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1C AC 

-h co co i— o 

1C r-t CO O CO 

AC rH GO CO' 

— 50 CO HN 
O OO r-t ic CO 

SD 05 CO rH SD 

CO CM AC O GO* 
O? th o O AC 

SD AO O'/ t- 

o: CO O 05 

10 rn t-h 


rH 05 SD lO 
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SD 05 tH t-h O 
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r Jl 


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rH rH ©? CJ 

CO CO rH rH 1 C 

SO i> GO 05 05 

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t-h t-h rH rH 



rH th rH 

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CO CO rH rH 1 C 

AO AO AO 

iOOO^N 

aC aC 

GO GO 05 05 O 

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lC lO 

• • 

1C 1C 
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CO CO rH th aO 


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CO rH SO l— 

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05 O CM CO AO 

r-i T—t T“H T —1 

AC O AC O AC 
to 00 05 — O? 

r-rnr^W 

OaOOaCO 
H- 1 C ( - GOO 
WWW wee 

lCOiOO 
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CO CO CO CO 


O O O O 
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GO *M 05 00 rn 
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SD O rr r- 05 

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T-h Ol ^ 05 

o o o o 

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63 
















































































TABLE IV,—D EFLECTIONS, Zero on first tangent. 
20-foot ChordjTper 2° .— Continued . 


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C5 X to Ci 

X X © 02 X 

X 

g. 


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0 


44 T—< 1-4 1—1 1-4 

w w ci oi w 

X 



02 X X 02 O 

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02 



—4 TP tp 1 —1 0 

44 TP Tp T-4 O 

1-4 



0 





05 0 ? CO O 

rt X CO C5 lO 

1-4 




W^COCOrJ' 

to 


tO 





r —1 








Tp 


d 



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-v 





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0 

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1-4 ri 1-1 

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64 







































































































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154.5 

162.4 



05 

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102.3 

106.6 

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rH CO 40 

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s-5 

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05 

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65 










































































































TABLE IV.—DEFLECTIONS. Zero on first tangent. 

5-foot Chords per 8°. 


tXT3 

G 

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O lO O lO OiOCi-rQ 
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TP CO 

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co co 


AC TP 


CO 05 

co co 


© X CO OP 


Tp CO CO 
O r-i CO 

o 

C? ~P CO 
Of Of Of 


CO 05 
O CO 


05 t-h 

Of CO 


co ao 
cp *-■ 


tp i> 
co co 


05 CO O O CO 

a: co Of co l- ol 10 —** tp 

CP t-H t— i t-h ■'P t-h lO CO 


TP CO QOOW’tN 05 C> 

t— th ^ Of Of Of Of Of CO 


05 CO CO X O GO 


COC>T<»t T- T-l 

OO WO C-CMO 


CO 05 t-h co TP CO GO 


CO Of 
© Of 


~ CO 
Of Of 


© TP 

AO cp 


AO GO 
CP CP 


05 CO CO CO i>- i.- ini'* 


Tp 

O 

o 

TP 


Tp CP X CP X X ~ i- 
OT^OJi O AT. TP 


O — 

ao o 


O co 

CP TP 


AO CO i> X O CP X iO t> © Of TP 

h, OP CP 



Of 

cl 

© op co 

t— AC X 

05 X 

00 GO CO iO 

Of OP CO AO y—t 

AO CO GO AO 

• • « • • 

t- ^ TP 05 O* 
■’p O? CO y~~* 

22.7 

40.3 


b 


CO TP iO CO GO 

05 ^ CO AO 

05 — 
T OP 


o 

of 

GD CO TP O 
« CO O TT 

05 GO CO 

• • • 

^P CO lO CO 05 
Of ^ ^ Ci CO 

cot-oxw 

cc -t o? o o 

CW-iOiO 

o o 

l‘ lO 

o o 


X 

0 

t—i r—* 

Of CO ^P O CO 

00 05 rH CO Tp 

CO 05 

»—■ T—■ 

• 

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—. -i-J 

OPh 

Of CO Tp iO 

CO t'- 00 05 o 

T-H O? CO Tp lO 

»—< t—< r- - y —H »-H 

co 

ri r—> 


© ao © ao 
— — o> o 

O lO o o o 

CO CO TP •*? »o 

iDOiOOiO 
AC CO *0 c- 

O to 

GO GC 

fe, 

—« -P 05 CO 

© © © tH 

O? O Tp co »o 

CO VO C- O Tp 

OP 05 ao O f o 
05 Tp T— 05 GO 

05 05 
i.~ X 


»—< y~~* 

^ OP CO CO Tp 

ao co 

C) 

0 

CO Tp OP o 

th OP X Tp 

GO CO Tp OJ O 
Tp iO co i> 00 

GO CO Tp OP O 
00 05 o — Of 

t— i r—« r—i 

CC CO 
Of CC 

TT T—• 


o 

:o 


w 

Oh 

cc 

Q 

Oh 

o 

w 

o 

H 

o 

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lO 



AO AO 

AO AO 

AO 05 Tp 00 CO 

CO O 


AO i- O Of 

AO l'- O CP »o 

NC5WTJ- 

05 CP 


T—' T—H 

T—1 T—1 CP CP CP 

CP CP CO CO CO 

X Tp 


TP TP CO O 

O TP Tp CO O 

CO Tp Tp CO O 

CO Tt* 


CP AO CO CO 

CO AO CP o o 

O CP AO CO CO 

X AO 

1 

rH Of 

CO Tp CO X o 

CP Tp CO 05 CP 

AO X 



T-H 

T-l T—1 TH r-H CP 

CP CP 




X 









Tp 

b 




1—1 

a 




l> 





CP 









CP 

CP co 

CL 



o 

X © 

b 



rH 

CP TP 




CP 

CP CP 

co 








X CO X 

Tp Tp 

Ph 



COO ri 

TP r-H 

b 



AO CO X 

© t-H 




T-l T- 1 T-l 

T— CP 





© 

aO 




• 



/ 

C? X X CP o 

of 

Ph* 


( 

O O rH CO AO 

rH X 

O 



tH cp CO Tp AO 

i>x 




rH y—t » — ' tH *—• 

T—l T— ( 




05 

© X 

TP 



• 

• • 



Tp on 

CO X Tp Tp { — 

AO t- 

Ph’ 


T—« O 

O O rn CP CO 

AO TH 

b 


7 

8 

C5 o H CP CO 

Tp CO 




^r-riH 

rH T—l 




C5 05 X 

L- CO 




• • • 

• • 



TP co CP CP 

CO <*P AO TH rH 

iO X 

p^ 


T-I AO Tp CO 

CP Of Of CO TP 

AO r —1 

b 


Tp Tp AO CO 

*> X 05 o T-l 

CP TP 




05 05 X L- 

CO X 

OP 






Of 

C> CO -p CO CP 

CP lO CO AO ^h 

H AO 

CL 

o 

CO O Tp CP rH 

O aO aO aO O 

TH CP 

b 

CP 

CP CO CO Tp AO 

CO CO i>- X o 

T-H CP 




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T—l T—H 




05 05 X CO CO 

X T“-» 

rH 



• • * • 

• • 


X CO X 

tp tp on co X 

CO CO tr JO 

X X 

Ph 

COaOth 

Tp T-l Tp CP O 

ao TP co CO X 

Tp AO 

b 

tH 

t—< cp op co tp 

Tp io co x 

© © 





»-H 




05 X X CO Tp 

TH 


X X c> o 

C? X X c? o 

TH — 05 

o © 


T—i X aO 

r—1 CO O Tp CP 

O Tp CO X CP 

X X 



•TH TH CP CP CO 

Tt Tp IO O N 

X © 


AO AO 

AO AO 

AO AO AO 

AO 


• • 

• • 

• • • 

• 


rH rH w W 

CO CO Tp TP AO 

AO CO CO L- 

X X 


AO AO 

AO AO 

AO ao ao 

AO 


♦ * 

• • 

• • 1 

• 


ao i - o cp 

AO N O W AO 

NOO)iON 

© CP 


y~~i t— i 

H CP Of CP 

CP X X X X 

Tp Tp 


O — C? -p 

X CP C5 CO CO 

X X O 05 Of 

X i- 


o o o o 

O T-l rH Cp CO 

Tp o X 05 Of 

TP L-- 




rH 

TH tH 


COTpOIO X CD TP CP © GO CO Tf w O QO CO 
r^WCOTp ^AOCOi.'-aO OOOOr^W OP X 


66 





















































































TABLE V.—RADII, AND DEFLECTION ANGLES IN MINUTES PER FOOT OF LENGTH. 


• 

«W % O 

io i o ao 

O lH *-H P> Pi 

tO 50 tO tO TO 50 

40 40 

CO CO -H *0 

to to to to to 

40 40 40 

40 to to N l- 
to to to to to 

40 40 

GO GO 05 05 

to to to to i> 

40 40 40 

O ^H TH Pi Pi 

L- L- i- t- 

40 40 

CO CO rf rf 40 
iN i- l- L- 

tf) 

3 

O CO O ' iO Tt 1 

t'- 05 TO Pi Pi 
40 O 05 — to 

CO lO *> QC 
'pi lO 05 to to 

pi O — 05 Pi 
05 iO CO CO i> 

OCOOOr- 
CO 1 — Pi 40 o 

05 GO i'- GO O 

1 - 05 CO o 

*30$ 

ce 

Ph 

iO rH GO »C ^ O 
TO rf i-H 05 l>- iO 
GO 00 00 i> i>- 
Pi Pi Pi Pi pi Pi 

GO t- iO »0 rf 
Pi O GO TO -n 
i- i- to to to 
Pi Pi Pi Pi PI 

Tf rf Tf 40 to 
Pi O GO tO rf 
tO tO 40 40 40 
Pi Pi Pi Pi Pi 

{— 05 rH CO 40 
Pi © 05 t- *0 
40 40 -f P rf 
Pi Pi Pi Pi Pi 

GO *— rf i •• *—i 
COCiOXN 
P P p CO CO 
Pi Pi Pi Pi Pi 

—’ GO Pi ib pi 

40 CO P> O 05 

CO CO CO CO Pi 

Pi Pi Pi Pi Pi 

bb 

OrH WCO 

to GO 05 O 

rH 

▼H Pi CO 'P 40 

rH rH i—1 i—t th 

to i> oo 05 o 

rH t— 1 i—i -rH Pi 

rH Pi CO rf lO 
Pi Pi Pi Pi Pi 

CO N GO 05 O 

Pi Pi Pi Pi CO 

Q 

Pi 






Def. 

per 

Foot. 

aO iO 40 

lO iG tO N N 

Tf rf if rf rf rf 

40 40 

GO GO 05 05 
rf Tf Tf Tf 40 

40 40 ' ' 40 

O 1 — —• Pi Pi 
40 40 40 40 40 

lO 40 

CO CO Tf rf 40 
40 40 40 40 40 

40 40 40 

40 to CO N N 
40 40 lO 40 40 

lO 40 

GO GO 05 05 

40 40 40 40 to 

W 

Zj 

O ' »0 GO ’’H’ tO 
O O OH' ^ G5 

40 Tf i — CO O 
Pi TO !—• c- O 

to 05 to ?> 05 
C5 40 GO C- Pi 

O 05 CO CO 40 
rf © CO r—' rf 

O P GO 05 tO 
CO tO rf i- 40 

05 to to GC O 
i- rf O O O 

*'3 02 

Ph 

O X to tO N X 
Pi *> CO 05 40 ^h 
GO t- TO to tO 

CO op TO CO TO 70 

-h rf» GO Pi 00 
GO P O N TG 
40 »0 *0 rf ~f 
CO CO CO CO CO 

COONOP 
Oi-WON 
Hi CO CO CO P> 
CO CO CO co CO 

CO CO co p 40 
-r — GO *P Pi 
Pi Pi !—<’—• *—i 
CO CO CO co CO 

i- 05 Pi 40 05 
05 to T — GO 
O O O O 05 
CO CO CO CO Pi 

CO GO CO 05 40 
tO CO »—• GO tO 

05 05 05 GO GO 

Pi 7i Pi Pi pi 

Deg. 

D 

O o> CO ^ »o 

CO CO TO CO CO CO 

to GO 05 o 

CO CO CO CO rf 

1—1 pi CO rp 40 
p pppp 

to i> GO 05 o 
P P P P 40 

rH Pi CO rf 40 
40 40 40 40 40 

tO N- GC 05 O 

40 40 40 40 TO 

o 

1—1 







Def. 

per 

Foot. 

lO 40 o 

o — p> Pi 

CO CO TO CO TO CO 

40 40 

CO TO Tf ^ 40 
CO CO TO CO CO 

lO 40 40 

40 to to N l- 
CO CO CO CO CO 

40 40 

GO GO 05 05 

CO CO CO CO rf 

40 40 40 

O rH rH Pi Pi 
P P P P P 

40 40 

CO TO Tf rf 40 

P P P P Tf 

<X 

D 

o o to o to CO 
O O i-h i—i GO Pi 

05 40 GO — CO 
O CO GO TO Tf 

to O GO rf O 
Pi O 40 05 o 

05 40 o o o 
to 05 C- 05 40 

40 GO GO tO O 
rf tO i—i GO i> 

QC CO Pi Pi O 
tO i- GO 05 O 

'S 0} 

c$ 

tf 

O to lO N H O 
CO TO -T' id < - GO 
i- to 40 rf CO Pi 
OOOIOOO 

05 —' »0 Pi *■“* 
O CO iO GO —• 
Pi — O 05 05 
OOiOPP 

Pi 40 05 40 -f 
PUOPX 
GO O in tO iO 
P P P P P 

O? H- r ?> 

Pi tO O 40 05 
40 rf rf TO Pi 
P P P P P 

H* Pi Pi Pi rf 
pi 05 rf 05 -r 
UirnOO 
rf rf rf rf rr 

t- TH TO Pi O 

05 lO O to Pi 

05 05 05 GC GO 

CO CO CO CO CO 

tiu 

Q 

© T-i Pi CO rf 40 

o 

rH 

to J >00 05 o 

T—1 

rH Pi CO rf iO 
i —1 tH i —1 rH i—i 

% 

to l> 00 05 o 

rirlnrCi 

rH Pi CO Tf 40 
Pi pi Pi pi Pi 

to 00 05 o 

Pi Pi Pi Pi co 


Def. 

per 

Foot. 

lO 40 40 

»G lO P to (>• N 

rH 1—1 1—1 1—1 1—1 1—1 

40 40 

GO GO 05 05 
i-i ^ ^ i-. Pi 

40 40 40 

O rH H p? C» 

Pi Pi Pi Pi Pi 

40 40 

CO CO rf rf 40 
Pi Pi Pi Pi Pi 

40 40 40 

40 to to N- t- 
Pi Pi Pi Pi Pi 

40 40 

GO OO 05 05 

Pi Pi Pi Pi CO 

CG 

to 

O CO 00 Pi GO GO 

O 05 GO o o 
O GO TO Tf O 

N h iO iO O 
CO N- CO to o 

H 05 O CO 00 
05 GO 40 CO O 

QC rf 05 O 

h iO N CD 05 

05 00 05 Pi o 

Pi 40 40 ~H O 

tf 

Ph 

© O CO GO H Pi 

to 05 TT — Pi 

O {- Tf ^ CO 
t— i i—i O © O’ 05 
rH tH t —1 i—i rH 

O H t- IO «o 

jO 05 "Hi i—i 05 
40 Pi O 00 40 
05 05 05 GO GO 

40 40 40 CO O 
GO GO 05 — 1 Tf 
CO rH 05 GO to 
GO GO 1- i.- iN 

CO rf Pi to td 
{- TH to -l- 
P TO rx O 00 
in i- c- L- tO 

r- 1 i—i to to o 

p rn zr to o 

N CO P CO C? 
ZO ZD TO tO tO 

05 i— < N- N- O 

CO — Pi P i co 

H o O X N 
to to 40 40 40 

&c_ 

® Cl 

Q 

© H Pi CO rf JO 
CO CO CO CO CO co 

o 

o 

tO O 00 05 O 
CO CO CO CO rf 

•rH Pi CO rf 40 
PPPPP 

CONXOSO 
P P P P 40 

rH Pi CO rf 

40 40 40 40 40 

CONX050 

40 40 iO 40 to 

i 

Def. 

per 

Foot. 

lO iO 40 

© rH T—< Pi Pi 

o o o o o 

40 40 

CO CO Tf TP 40 

o o o o o 

40 40 40 

40t0t0N i>- 

o o o o o 

lO 40 

GO GO 05 C5 O 

O O O O TH 

40 40 40 

O rH rH Pi Pi 

rH H l-H 1—1 1—1 

40 iO 

CO CO Tf Tf 40 

1—1 1—1 th 1—1 1—1 

rf) 

o o o o o 

O co o o o 

ioo:>ho 

40 40 O N O 

Tf Pi GO o O 

rH to to H O 

Radiu! 

R 

©©©>©© 
OOO'OtO 
CO 05 to 05 t 

co i-i -r 40 go 
—r — QO to 

CO i—i r—i 

O rfi 40 O O 
O i—• i'- O GO 
CO i—< 05 Pi TO 
L- 05 Pi r -h 

40 p p t: to 

t © to < - © 

40 40 H* iO Pi 
Pi to P 40 05 
rr CO to P Pi 
CO Pi Pi Pi Pi 

i— TO O P o 

GO Pi O 05 05 

rf 04 —« O — 

HGC5 00N 
Pi Pi rH r-* rH 

— i- i- ib Pi 
L- Pi P Pi 40 
CO TO 05 CO i'- 
to lO -r H CO 

t— i i—i rl i—i i-H 

CO CO GO ib o 

Pi TO i- 40 to 

Pi i> Pi CO rf 

CO Pi Pi rH l-H 

rH i—i rH rH rH 

Deg. 

D 

O rH Pi CO ^ 40 

O 

O 

TO N- CO 05 © 

rH 

i— ' Pi CO rf *0 

iH f —1 r—l T—1 1—1 

O NX 05 O 

r-i - i—i i— ■ Pi 

r- Pi CO P 40 
Pi Pi Pi Pi Pi 

tO l> GO 05 O 

Pi Pi Pi Pi CO 


67 











































































TABLE V—RADII, AND DEFLECTION ANGLES IN MINUTES PER FOOT OF LENGTH. 


«w h o 

* io 

p 

tO tO tO 

CO CO CO CO CO CO 

t—> t— 4 r—< t—1 t-H t—1 

tO iO 

GO GO 05 02 

CO CO CO CO Tp 

r—1 t—4 T-H 7—< t—4 

1.405 

1.41 

1.415 

1.42 

1.425 

to to 

CO CO ^4 rr tO 
TT 1 TT Tf ^ ^ 

• • • • • 

T—< 7—4 7-4 7—4 7—1 

to to to 

to CD CD t- 

TT TP ’’t TP TT 

7—1 T-H t— 1 r—1 r— 

to to 

GO GO C5 05 

rf Tf TP tO 

r4 7—4 7—4 7-^ 7—4 

Radius 

R 

CO co TP ^p GO 

CO CO 05 CO O 

03 X CO 05 Tf O 
i' O CD lO O O 
03 03 03 03 03 03 

t—t T—1 T—« T—1 *—1 T—1 

CD CD O CD CD 
CD t-. L- 03 GO 

tO T—i CD 03 O 
Tp tt GO CO 03 
03 03 73 73 73 

7—1 7—1 7—1 T—1 7—7 

1223.49 

1219.15 

1215.05 

1210.56 

1206.31 

1202.10 

1197.91 

1193.75 

1189.62 

1185.52 

1181.45 

1177.40 

1173.38 

1169.39 

1165.43 

1161.48 

1157 57 

1153.69 

1149.83 

1146.00 

<D f-H 

o - o) co Tf io 

CO CO CO CO CO CO 

CDNG0 050 
CO CO CO CO TP 

W 73 CO Tt tO 
Tp Tp ''tf' TP TP 

CO J> aO 05 O 

Tf TJ4 TT to 

03 CO tO 

to to tO tO lO 

CD N GC 05 O 
to to to to CD 

Q 

o 

TP 






ni 

tO kO »o 

OH-03 0 
03 03 03 03 03 03 

to to 

CO CO TP TP to 
03 03 03 03 03 

to to to 

to O O i> N 
03 03 03 03 03 

to to 

GO GO 05 05 

03 03 03 03 CO 

to to tO 
O th 7—i 03 G^3 
CO CO CO CO CO 

to to 

CO 70 tP tO 

co co co co co 


rtrlrHHnn 

T—1 rH 1—i 7—( 7—t 

rirtHHH 

TH 7—4 7—4 7-4 7-H 

7—4 7—1 TH 7—i 7—4 

7—1 7-H 7— 1—4 7—1 

03 

□ 

OiOCOWr^O 

lOOOXOOi 

CD O 05 03 O 
tO 05 03 l- 03 

03 05 O tO CO 
{> 03 05 to 03 

CO Tf CD 03 7—4 
05 t- to ^ oo 

Tt 7-H CO {> lO 
03 03 0> 03 CO 

GO TP rr {> CO 

TP CD GO O CO 

l 05 

P3 

03 CD O 05 CO 

CO 03 O) rn O O 

TT Tp Tp Tp Tp ’■'P 
7—• 7—1 rH 7-4 7—1 »—1 

{'.i-CDO tO 
05 05) GO 00 £ '• 
CO CO CO CO CO 

rH 7—i *—i 7—i 7—i 

05 rr GO CO GO 
O C 17 »Q 'I' 
CO CO CO CO CO 

7—1 7—1 7—4 7—1 7—1 

0> {> 03 J> 03 
Tt 1 00 CO 03 03 
CO CO CO CO CO 

7—i 7—4 7-4 7-4 7—1 

03 {> 03 l> 
7— 7- o O 05 
CO CO CO CO 03 

7—1 7—1 TTt 7—1 7—4 

OM- 03 GO CO 

05 GO GC {.'• t— 

03 03 03 C3 0x3 

7-H t-H 7—1 7—4 7-H 

Deg. 

D 

O H o? CO ^ iO 

o 

TP 

CO L- GC 05 O 

7—1 

7-H 03 CO TT to 

7—1 7—1 7—1 T—1 7—1 

CD GO 05 O 

T—1 7—4 7—1 7—1 03 

t-H 03 CO tO 

03 03 03 03 03 

CD L- GO 05 O 

03 03 03 03 CO 

'S'® I 

tO »o *o 
tO tO to CD i> <- 

o o o o o o 

to to 

GO GO C5> 05 

O O O O t-< 

»o to tO 
O 7— 7— 03 03 

7—1 7— < 7—1 7—1 TH 

to to 

CO CO Tt rt 1 to 

7—1 7—1 7— 7—1 7—1 

to to to 
to CD CD N l- 

7—1 7—1 7— rH 7—1 

to to 

GO GO 05 05 

7 - - T—1 T-H T—1 03 


»— 1 7—1 rH T”H 7—1 rH 

7—1 7—1 7—1 7—1 T—< 

7—1 7—1 7—< 7—1 7—1 

TH 7—1 7-4 7—4 7—1 

7-4 T—1 7—1 7—1 7-H 

7—4 7-H 7—1 7—4 7—1 

C/3 

IT. X O 05 N 

t— • CO i 1 " O tO O 

i- tP *> CO 

CDCOOXN 

CO to — 03 O 
CD CD i> GO O 

^ O 03 GO 

03 ID C. CO i> 

03 O rf CO GO 
CO 05 JO 03 05 

XCOtpOO 
i> CD tO tO tO 

aS 

Ph 

N05H-^CD05 
CO 03 03 ^ O 05 
CD CO CD CO CD *0 

7“ ' 7—1 7—1 »— * 7—1 r—1 

r- Tp i> 05 03 
C5 GO t- CD CD 
ID tO tO tO tO 

7—1 7—1 7—1 7—1 7—1 

tO GO 7—i GO 

tO — Tp CO 03 
tO to to *o to 

7—1 7—1 7—1 7—1 7—1 

th T}1 {- rH T 

03 rH o O 05 
to to to to 

7—4 7—1 7—1 7—1 7—1 

GO t- 1 tO 05 03 
GC GO i> CD CD 
'TT Tt rf 

TH 7—1 7— 7-4 7—4 

CD O Tp GO 03 
tO tO Tp CO CO 
”^P TT TP 

7-H 7— 7—1 7—4 7—1 

Deg. 

D 

O 03 CO Tp tO 

CO CO CO CO CO CO 

0 

CO 

CDNX050 
CO CO CO CO Tp 

7—4 03 CO Tp tO 
TP Tp Tp Tp ^P 

CD i> GO 05 o 
^ ^ ^ TT tO 

7-i 03 CO to 

to AC to to to 

CDNG0 050 
to tO to to CD 

Def. 

per 

Foot. 

tO tO to 

O r-HO?:» 
05 05 05 05 05 05 

to to 

CO CO -t tP tO 
05) 05 08 05 05 

to to to 

tO CD CD l> i — 
05 05 05 05 05 

to to 

00 GO C5 05 

05 05 05 05 

7—1 

to to to 
O 7-1 7-H 03 03 
O O O O O 

7-^ 7—4 T—» 7—1 TH 

to to 

CO 00 TP -r to 

O O O O O 

7—4 7—1 7—4 7—4 7—1 

CO 

O »C TH 05 00 GO 
C^OCDttCO 

05 O 03 to GO 
CO lO i> O 

O 03 to GO 
O CD CO 7-i O 

00 to CD CO o 
O T-H CO CD O 

to GO O O GO 
rr 05 CD CO O 

CO i- GO GO to 

05 GO CC 05 rH 

1* 

PS 

O C5 05 GO GO GO 
■*-* 05 CO L-- CD tO 
05 X X GO 00 GO 

1-1 t— i t—< t—1 t—■ t— t 

GO CO GO 05 05 
rf CO 73 — O 
CO GO GO GO GO 

7—1 7—1 7—1 T—i 7—1 

O O 7—i 03 CO 
O 05 (JJ t> CD 
GC i- i- i- i- 

7—1 7—1 7—1 T"H 7—1 

Tt to CD i> C5 
tO Tt' CO 0> t— i 
l- l- i- t- t- 

7—1 7“1 7—1 7—1 T—4 

O H CO 1C N 
7—i O 05 GO i— 

< — l— CD CD CD 

7—1 7—1 7—1 7—1 7— 

CO O 03 Tp t>- 
O CD tO Tp CO 

CD CD CD 90 CD 

7—1 7—1 7—1 7—1 7—1 

Oift 

ft 

OthWCO^O 

o 

CO 

CO i>00 050 

7—1 

7— 03 CO ^ tO 

7—1 7—1 7—4 7—1 7—4 

CD {> GO 05 O 

7—1 7—4 T-1 7—1 03 

03 CO tO 

03 03 03 03 03 

CD GO 05 O 

03 03 03 03 CO 

Def. 

per 

Foot. 

to to to 

»C »G cc CO N i- 
L- L- i- {> t.- c- 

to to 

GO GO 05 05 

L— L- i> L- GO 

to to to 

O 7-1 1— 0> 03 

GC GO GL GO GO 

to to 

CO CO r}4 Tf to 
GO GO 00 GO 00 

to to to 

to CD CD {> {> 
00 GO GO GO 00 

to to 

GO GO 05 05 

00 GO GO GO 05 

CO 

3 

OWiOCONO 
O CO CO O tP O 

tO 03 tO tO 

00 GO 05 03 {> 

t— 03 O Tf CO 
Xf 03 03 CO CD 

05 05 CO 03 tO 
O CD ^ CO CO 

CO 05 CD i> 

to 00 03 GO to 

thNONO 
XCO TTCDO 

•f—1 

CC 

Ph 

03 CD —* i> 0> GO 
cr. i> CD tP CO —1 
0> 03 03 03 03 03 
03 03 03 03 03 03 

CO 05 tO 03 GO 
O X t- o ^ 
03 —- — i—< 7—i 
03 03 03 03 03 

tO 03 05 CD CO 
CO 03 O C5 CO 
7- — — O O 
03 03 03 03 03 

7-4 GO CD 03 

tO -r CO 03 
O O O © © 
03 03 03 03 03 

O' GO C- tO t}4 
■7— 05 GC {.■— CD 
O 05 05 05 05 

03 7—1 7—1 7-4 .-4 

CO 03 7-4 o O 
to ^P CO 03 TH 

05 05 05 05 05 

7—1 7—1 7—1 7—1 7—4 

bi— 

oft 

ft 

O *—< 03 CO tO 

CO CO CO CO CO CO 

O l- GO 05 O 
CO CO CO CO "sT 

7-. 03 CO rt tO 

Tj4 TJ4 Tj1 Tjl Tf 

CD i> GO 05 O 
Tt Tji r^i tO 

7— 03 CO Tt to 

tO tO to to to 

CO N 00 05 o 
to to to to CD 

o 

03 







68 


M 








































































TABLE V.—RADII, AND DEFLECTION ANGLES IN MINUTES PER FOOT OF LENGTH. 


• 

^ *H O 

^ Jo 

lO iO lO 

O r* ■*—i ©* ?» 

tH rH rH 1—' tH rH 

40 40 

CO CO Th T iO 

T—i r—< tH rH t —1 

40 40 40 

lO CO CD N N 

rH »—« rr •—> rH 

40 40 

GO GO 05 05 

T-H rH rH rH ©? 

40 40 40 

O — TH ©) 

©? ©V* ©? ©1 ©l 

40 40 

co co Th Th 40 

©* ©^ ©* ©* ©i 

Q 

©* ©* ©* ©* ©* ©* 

©* ©* ©* ©* ©* 

©*©*©*©*©* 

©i ©* ©i ©i ©i 

©> ©i ©> ©i ©i 

©^ ©i oi ©i ©i 

m 

d 

•»h . , 

i> ©J 05 GO 40 Th 
CO O *sO l- X 05 

-f 40 *> O Tt» 
O rH Of rf 40 

GO^ONiO 
CO CO O t—« CO 

CO Th CO Hf CO 
40 i— 05 rH CO 

05 co t- CO GO 
40 C/D O CO 40 

40 CO rH o O 
XHTJI^O 

■S 05 

PS 

OOCD^OiOQO 

1—' 1—1 r-l rH T—' O 

00 OO GO 00 GO 00 

40 CO —* 05 
O O O O 35 
GO GO GO 00 L- 

i- 40 Th ©? O 
05 05 05 05 05 
in i.'- l> i- L- 

GO CO Th CO T-H 

GO 00 00 00 GO 

L- i - t- 

05 N CO Th ©? 

1- i- 
i- L^i-irN 

o’ 05 40 Th 

co CO co CO 

Lr i.- L- J> tr 

Deg. 

D 

O tH ©* CO Th 40 

o 

i> 

CO J>00 05 O 

■rH 

h ©J CO Th 40 

rH rH th t-h Y=i 

CO 00 05 0 

vH rH rH rH ©^ 

T-H CO Til 40 

©l ©* ©i ©i ©i 

CO N X 05 O 

©< ©i ©* ©1 co 

■s«| 

40 iO 40 

lO lO O 50 f- t'« 
05 05 05 05 05 05 

40 40 

GO GO 05 05 

05 05 05 05 

40 40 40 

OnHOlCI 

ooooo 

lO 40 

co co Th tp 40 
ooooo 

40 40 40 

40 CO CO t- t- 
OOOOO 

40 40 

GO CO 05 05 

O O O O rH 

• 

1-H 1—• rH T—( rH rH 

T=H HHriCi 

©i oi ©* ©* ©* 

©i ©i ©i ©i ©? 

©i ©1 ©? ©* ©i 

©i ©i ©j ©i 

CO 

3 

Th 00 Th O 00 OO 

ocjocooco 

go 05 cro o 

rH 05 00 CO 40 

CO ©? O 05 05 
CO Cl rH 05 QO 

O ©? *0 05 Th 
GO ir CO iO 40 

O Th co co 

40 Th Th Th Th 

ThC0 05CO£r 

Th Th TT 4C lO 

P3 

r-i 05 O Th ©* O 
00 I- {> l- {> c- 
00 00 GO 00 00 00 

GO 40 CO rH 05 

co co co co 40 

CO GO GO GO GO 

j> id co o cb 

40 40 40 40 Hf 

00 GO co GO GO 

CO Th G^ O GO 
Th Th Th Th CO 
00 GO GO GO CO 

CO* Th ©> O X 
CO CO CO CO ©* 
GO GO GO GO GO 

cd Th ©i o x 

©* ©?©?©? rH 

00 GO 00 GO 00 

&b_ 

Op ft 

ft 

O rH ©* CO Th 40 
CO CO CO CO CO CO 

o 

CO 

50NG0 050 
CO CO CO CO Th 

rH 0$ CO Th 40 
Th Th Th Th Th 

CO NGO 05 O 

Th Th Th Th 40 

rH ©* CO Th 40 
40 40 40 40 40 

ONX050 

40 40 40 40 CO 

^11 

40 40 40 

O — rH 0* 01 
00 00 00 GO GO 00 

40 40 

CO CO Th Th 40 
00 GO GO GO GO 

40 40 40 

40 CO co b- l> 
00 GO 00 GO 00 

40 40 

GO CO 05 05 

00 GO CO 00 05 

lO 40 40 

O H TH o? ©^ 

C5 C5 05 05 05 

40 40 

CO CO Th rh lO 

05 05 05 05 05 

«°l2 

r— > rH t-h t-h t— ( *—* 

rH r—i t—1 t-H rH 

rH rH *h rH rH 

rH rH rH rH rH 

rH rH rH rH rH 

tH rH rH rH rH 

CO 

OiOCO’HH'T? 
O CO ^ lO o 

40 05 rf —‘ 05 
CO t- Cl lr rH 

05 O ©J 40 O 
CO ©? l> ©i 00 

CO Th ©? CO Th 
CO 05 lO T—i i> 

CO O 40 rH 05 
CO O co CO 05 

£— t>- GO O Th 

CO CO O 00 40 

i 0 * 

PS 

lO ©i 05 Th —1 

o lO *t rT 1 ^ 

05 C5 05 05 05 05 

05 CO rH rH 05 
CO CO CO CO ©* 
05 05 05 05 05 

CO Th rH 05 CO 

©? ©J ©* rH H 

05 05 05 05 05 

Th rH 05 i- -H 
rH r-i O O O 

C5 05 05 05 05 

d o n »o d 

! 0 ) 05 05 05 
05 05 00 00 GO 

O GO CO CO rH 

05 GO GO X QO 

CX GO GO GO 00 

&L 

0)ft 

ft 

br-iWcoNt'o 

o 

co 

CO l> GO 05 O 

rH 

rH CO 40 

H r-i rH t-H rH 

CDNG0 050 

TH rH rH rH ©i 

tH O? CO Th 40 
©i ©i©i ©^ ©^ 

C0NCCC5O 
©*©*©*©* CO 

Def. 

per 

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40 40 40 

i0 40?OCNi- 
CO CO CO CO CO CO 

rH i— i rH tH T—l 1—1 

40 40 

GO GO 05 05 

CO CO CO CO £> 

rH rH rH rH rH 

lO 40 40 

Ohh^'M 
i> i> fc- i> 1> 

rH rH rH rH rH 

1.73 
1.735 

1.74 
1.745 

1.75 

40 40 40 

40 CO CO It tr 

tr i> t> t- 

rH rH rH rH rH 

1.78 
1.785 

1.79 
1.795 
1.8 

Radius 

R 

©* X Th CO tt t> 
00 CO 40 Th co o? 

t-h GO 40 ©? 05 CD 
Th CO CO CO ©* ©* 

o o o o o o 

rlHnnriH 

tHGOCOON 
Gi H H H H 

CO O N Tf H 
ft> O* rH rH H 

o o o o o 

rH rH rH rH rH 

1008.21 

1005.26 

1002.33 

999.42 

996.53 

40 05 OD O 05 
CO i- 05 rH ©* 

co d n io ci 

05 05 00 GO CO 
05 05 05 05 05 

05 Th 05 40 

Th L— 05 rH Th 
• • • ♦ • 

05 CO CO rH GO 
*> l- l- zD 

05 05 05 05 05 

co co Th »o o 

N O CO CO O 

40 * cd o* i- 40 

CO CO CO 40 40 

05 05 05 05 05 

si- 

CD ft 
ft 

O rH ©* CO Th lO 
CO CO CO CO CO CO 

o 

40 

CD N GO 05 O 
CO CO CO CO ^ 

rH 'N? CO Tf< lO 
Tf Th Th Tf Th 

CO O GO 05 O 
^ Th Th Th 40 

rH ©* CO Th 40 
40 40 40 40 40 

CO i>GO 05 O 

40 40 40 40 CO 

<n X> © 

lO *0 40 

o — H 0* CJ 
lO 40 40 tO 40 40 

40 40 

CO CO Tf T »o 

40 40 40 40 40 

lO 40 *o 

4C CO CO ^'-IT- 
40 40 lO O 40 

40 40 

GO GO 05 05 O 
40 40 40 40 CO 

40 40 40 

OhhOICI 
CO CO CO CO CO 

40 40 

co co t rf io 
co co co co co 

ft°£ 

thhhhhh 

rH rH rH rH rH 

rH rH rH H rH 

rH rH rH rH rH 

rH rH rH rH rH 

th rH rH rH rH 

CO 

O 05 H 50 W H 
o "T 50 05 C? 

CO ^ CO 0 ? CO 
40 GO ©? CO O 

CO ©* O —« CO 
Th 05 rf 05 

t- -T CO Th CO 

05 40 r-. i.- CO 

CO O O rH Th 

O L- Th rH GO 

O X Lr 05 ©> 

CO CO r 05 x 

d 

cc 

ft 

CO CJ X) rt< O N 
rf *f CO CO CO c? 

t—< •—i ■—• r—l rH rH 
1-H rH t—1 t-H rH rH 

CO 05 CO O? 05 
ClrHHH O 
rH rH rH r-• rH 
rH rH rH rH rH 

40 rH GO Hf* rH 
O O 05 05 05 

rH rH CD 

TH tH t-h rH rH 

Th t— Th 

GO X; GO o-O- 
OOOOO 

rH rH TH rH rH 

rl N Th H i> 
co CO CO 40 

ooooo 

T-H rH rH rH rH 

Thr-GOTh — 

40 40 Tf T T< 
OOOOO 

rH rH rH rH rH 

&L. 

a; Q 

OrHCJCO^lO 

CO J> GO 05 O 

rH 

th wcorf iO 

rH rH rH rH rH 

CO *>GO 05 O 

rH rH rH rH C? 

rH ©* CO Th 40 

©* ©1 ©l ©1 ©1 

C0i>XC5O 
©* ©i ©> ©^ 00 

ft 

o 

40 







69 

































































RADII, AND DEFLECTION ANGLES IN MINUTES PER FOOT OF LENGTH. 


> 

H 

iJ 

M 

<1 


Def. 

per 

Foot. 

^ W CC lO ONCOC5 i—< 0> CO tP lO 5DNGOC5 r- CO rf O CD t— OC C5 

CO CO CO CO CO CO CO CO CO CO ^ Tt 1 tT »C 1C lO *C lO »0 ic 1C lC ICO 

CO CO CO CO CO CO cococococo oo CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO 

i 

Radius 

R 

520.91 

519.34 

517.77 

516.22 

514.67 

513.13 

511.61 

510.09 

508.58 

507.08 

505.59 

504.11 

502.63 

501.17 

499.71 

498.26 

496.82 

495.39 

493.97 

492.55 

491.14 

489.74 

488.35 

486.97 

485.59 

484.23 

482.87 

481.51 

480.17 

478.83 

477.50 

i 

Deg. 

D 

OW^OQOO W NH *sO CO O W^iOCOO WrJicDXO O? TP CD GO O O? TP CD CO O 

Q th T-iT-!r-iT--<o* o* o* o* o* co co co co co xr rr o icio *d ioco 

tH 

1—4 

l i 

Def. 

per 

Foot. 

▼HO^CO^lO CiNGOO rH^COTf iO ONCOC5 T-HOJCO^ftiO CD l- CC C5 

OOOOO OOOOtH r-i t-h t—< i—i tH t-h -t—i *—< O? O'? O? Oi Oi Oi Oi O? 07 CO 

CO CO CO CO CO CO cococococo cococococo COCOCOCOCO CO CO CO CO CO CO CO CO CO CO 

• 

1 

Radius 

R 

© O O CO CD O CD CO OJ th D> CO O o lO C? 05 N D N D O? ID © CD O* 

O ri C) CO Tji D £- C5 r— - CO O © O? JO 00 *-» lOXC? iCO CO^O tD 

MihOJNiOCO T-< C5 GO CO tP C? O D N lO CO C? O CO N lC CO C? C 00 i- lD Hp Oi © 

N t- C CD CO O CD lO ‘O o O lO 1C o O O O Tp CO CO cocococo o> o> o> ?* o> 

ID lO 1C lO IQ O AO AO O iD O lOiCiOiOlO AO AO AO AO aO lCiOOiOiO lO lO iO C ib 

6ib_ 

Q 

OCJ^OQOO 0$ TP © CO © CiTpCDOOO C? TP CD GO O O* 'rp CD CO © O Tp CD OO O 

Q *“» T-inriH^ 0^ O-* O? C\£ CO CO CO CO CO ^ Tfi Tt 1 TP O ID lOiOOCD 

O 

rH 

1 l 

Def. 

per 

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. t-H D> CO 'TP AO ONGOD ^ ^O?C0^iD ONCOC5 t-i O? CO TMD © QC C5 

i> i- i- t- i> i- i-i.-t-i.-GO GO GO 00 GO GO CO CO QD GO 05 D D CT. O. D SciDO 

O* O* 0* O* 0* O* O* ©* 0* OI O* O* OJ C* ci oi oi Oi ci Q* 3* 09 d Oi ci oi oi C? CO 

Radius 

R 

i> 52 05 SO i> 05 CO GO AO CO 'Tf t- GQ CO © lO lC CO ^ D C) O D C5 rH tP D lO C) O 

CD CO 05 CD CO O OOiOCOr-D 1 -iO^Nh © 05 00 00 £- t- t- © © i- t- t - 00 05 © 

^ It ^ 55 40 2? O °0 © CO »— 05 i— ID CO i— 00 © tP 0i O GO © tP oi O X CD rP GO* 

CO CO CO D? D? D* W74- i-" — -r-OOOO © 05 05 05 05 DQOXX^ 00 <-?-£•- t- 

5DCDCOCOOCD OCOOCDO COOCDCDO CDiOiOiCO lOiDcS-C ID aO lo aO lb 


OW^CDXO WtHOXO ^ CD 00 O C* -p © CO © CJ^CDCOO W-fCDXO 

»—< rir-riT-iOJ C^OiG^OiCC CO CO CO CO Tf Tp tj* ^ o IQ lD lO iO CD 

C5 

Def. 

per 

Foot. 

rHt^COrPO © £— GO 05 © t- GO © -rH C? CO tP )P CDt-QOCi 

rPTpTP^'TpTl' ^ T-p TflD 1C O O id ID iO IO ID IO O CD © © © © D CD CDCD N 

oioioi oi oi oi oi o* oi oi oi oi oi oi oi oi oi oi o* o* oi oi oi ot oi oi ci oi oi o? o» 

Radius 

R 

ID GO CO r-H —i CO X ID t# CO O © TP aO £- O* 00 t- 00 *h AO ^ ro c\> -f v ~ 

W Ci CO TP ID O t- 05 1-H CO CD CO H TPi- r, TpX^Nrn 0? 00 -Tp © © ' 

O W O i- h COiDCOOi- TP 0> 05 CD tP — CO © CO t-- GO CD CO -r-^ 00 CD CO « rji cr 

r— « r— ( O O O 05 05 05 05 DO GOQ0i.-i.-t- NDOCO AD ID *D Jo tP 

t- t- t- i> L- t- CD CD 5D CO CD CD CD CD CD CD COCOCDCDCO © © © © © OoSoS 

Deg. 

D 

© 0* tP © 00 © O? TP © 00 © O? tP © CO © O* ^P © 00 © ^ TP CD 00 O D) rh co m r, 

o rH o?c^(^^co cococo oiSooS 

GO 

Def. 

per 

Foot. 

lO 40 lD _ *D ^_ ID iD ID ID ID lD lD in iD ,a 

iD *D CD CD t— i'— 00 GO 05 05 O — — O? 0* COTOtPHiD lDCDCDt-t- CDcn^rZ 

o?o?Gv? 0^0? Di o? o? O/ CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO TP 

ei oi oi oi oi oi oi oi oi oi oi oi oi oi oi oi oi oi oi oi oi o? oj oi oi oi oi oi o? oi oi 

i 

Radius 

R 

O O? t- O ID O CD O? C5 t- CD ID lD CD t— C5 O? lD C5 H m D) h v/-s 

5««00i?J?0 OS CO O O CO 1-r-iCOiM Nr.®Oi< OM®M® 

"VP O? O GO t— iD CO D? O C5 i- ID rp 0? O C5 i- CD rp CO th C35* cri eri rn >-.7 * D 

ggggee ggg^g 

Deg. 

D 

«W«W° SSSSS toSSSS 

o 

t- 


70 























































































TABLE V.—RADII, AND DEFLECTION ANGLES IN MINUTES PER FOOT OF LENGTH. 


Def. 

per 

Foot. 

in in in 

CO CO Tt Tt 1 C ic 

CO CO CO CO CO CO 

1C ic 

CO CO i> i> GO 
• • • • 
CO CO CO CO CO 

IO ic lO 
00 05 05 o 

cd cd co* i> i> 

io m 

T~i T-i Of Of CO 

id id id id t- 

m m m 
coTp^mm 

id id id id id 

ao in 
to CO i> i> GO 

id id id id id 

Radius 

R 

CO h 05 ri CO ^ 
00 i> iC 1C rt rt 

C? O GO CO rt* Ol 

i- i.>- co co co co 
Oi Ol Ol Ol CM Ol 

ic o N N O 

TT lC ic CO 

o go co* rr oi 
CO iO ic 1C 1 C 
Ol Ol Of Of Ol 

iO CO rf N CO 
05 r —1 CO lC GO 

o 05 1 >- io cd 

iO rf rf rf rr* 
Oi CM <0* O? 

^-ouoox 

r- Tp r-. TP 

CICXNiO 
Tf Tp’ CO CO CO 
G^ Of Of Of Of 

00 O Tp o 00 
GO CO i.^ 0* CO 

cd o? o’ 05 id 

CO CO CO Of Of 
Of Of Of Of Of 

CC in n 00 
t—' i> Ol 00 CO 

cd tj! cd 7-i* o 

Oi Ol Ol Ol Ol 

Ol Ol Ol Ol 01 


o o o o o o 

i-' Ol CO Tt lO 
o 

o o o o o 

T—1 <0* CO rt 

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lO ri CO 

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CO Tj< AO r—* 

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ft 

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CO 

24 

in 

Ol 



ic 1C ic 

GO GO 05 05 o 

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•H TH c? W CO 

iO IO iO 
CO Tf iO O 

m m 

CO CO fc- L- GO 

m ao in 

GC 05 05 o 

in m 

t— i Ol Ol CO 


^ ^ ^ ic o 

lOiOiClCiO 

IO IO lO IO iO 

m in m in o 

io m m co co 

co CO CO co o 

Radius 

R 

i?) CO 7) N O O 
T-' rt GO Ol GO ry 

co 05 ac rt -* 
O i- lO ^ CO 

■*— CO t-h lO co 
CO CO TP iC 

CO ^ GO O GO 
05 G^ O C5 CO 

io CO ^ o CO 
GC CO 05 AO r-t 

O 7) CO Tp CO 

GO in Ol O GO 

00 ^ o ^ w o 

O iO »C Tf Tf 

CO CO CO CO CO CO 

^ CO 6 N Tf 
CO CO CO Ol 01 
CO CO CO 00 CO 

th qo id oi 05 

Ol -—' o 

CO CO CO CO co 

CO rt r- CO co 
O O O C5 C5 
CO CO CO Of Of 

CO i—i CO CO 

05 05 00 GC GO 

m gi of of of 

■»— C5 fc- in O! 

GO i- c- i- t- 
Ol Cl Ol 01 01 

60 

© o © o o o 

y— 1 Ol CO rr 1C 
o 

o o o o o 

■H Ol CO rt 

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lO T-t C* CO 

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TF io -rr Of 

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co Tf in t-< 

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oi co xt* in co 

o 

CO 

7—1 

i>* 

T—I 

00 

1—< 

C5 

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s 


Def. 

per 

Foot. 

Of CO iC 

ic ic 1 C 1 C ic o 

Tt rt Tt TP rt Tt 

CO i> GO 05 

ic AC lC ic CO 

rr Tt Tf 

1-* O* CO Tt lO 
CO CO CO cc CO 

TP 1 Tf Tjl Tt Ttl 

CONXC5 

co co co co 

Tf Tt< Tf Ttl Tf 

th G? CO ’Tt' lO 
i> C- t- c- 

Ttl rjl Tf TT 

CO 00 05 

i> i> i> i> go 

rf Ty Tf Tf Tf 

Radius 

R 

O ic - N ^ o 

Oth??tJi CO 30 

N1CCO-HC5 

05 1 — co ic co 

C5XXNX 
00 O Of Tf CO 

c: C5 —i co An 
qo o co m t— 

i^ o CO CO o 

05 O! ^ CO 05 

CO 00 01 i> 01 
i— 1 CO CO 00 *-i 

Ol —• O 05 od i- 
GO GO 00 i'- i— 

CO CO CO CO CO CO 

CO CO lC rt CO 
i- t - i><- 

CO CO co CO CO 

WGIhOO 
1- t- c- CO 

CO CO CO CO CO 

GC GO CO id 

CO CO CO CO CO 
CO CO CO CO CO 

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to co co co co 
co co CO co co 

n005 00 GO* 

CO CO AC AO AO 

CO CO CO CO CO 

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r-1 t—I 7—i t—I Ol 

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Ol rf to 00 O 

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r* oi co -t ic 

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CO O GO 05 

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TH O? CO ^ lO 

co CO CO co co 

CO i- 00 05 

CO CO CO CO ^ 

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TT' TT TT 1 

CO 1>- 00 C5 

Tp Tp Tp ’T An 


rt rt rt ry ry ry 

rr Tt rr rr rr 

Tt' Tf TT 1 Tf Tf 

Tjl Tjl Tf Tf 

Tf TF Tf "Tf Ty 

rt rt rt ^ Ty 

Radius 

R 

05 — AO 05 70 i- 
Ol CO CO CO ry ry 

O GO ^ O N 
1C aC COl'-t'- 

^ C> o X N 
00 C5 o O rH 

t-— N CO N GO 
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CO 7? t~ o 05 
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rr rr rt rr co 

GC i- t- CO id 
05 05 05 05 05 
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CO co CO CO CO 

05 00 00 ir- CO 
GC GC 00 GC GO 
CO CO CO CO CO 

AO rt CO Of oi 

OC' GO 00 OC' 00 

CO CO CO CO CO 

b£ 

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05 05 05 05 05 05 

CO ^ CC 05 

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C N X C5 

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i—* oi co tt An 

i—• i—« 7—1 r—• 7—i 

co t— ao 05 

7—1 7—1 7—1 1—1 Ol 


CO CO CO CO CO CO 

CO CO CO CO Tf 

TT ^ ^ 

TT ^ ^ Tji TJ 1 

Tf T}’ rf Tf 

xt "rt ^ rt ry 

Radius 

R 

i> rt 0? O 05 05 
N O O Tt C? 

05 O »—< CC lO 
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GO — lO 05 lO 

co co *C Tf TP 

OOWON 
TP CO CO CO o? 

m co o> oi i-H 

01 Ol 01 01 Ol 

C> CO rt CO 05 

Ol Cl Ol Ol Ol 

O 05 GO i> CO ic 
—r TO CO co rc CO 
rt rt rt rr rr rr 

OCO-OC5 
CO CO CO CO 01 
^ orf o o 

GC co iO T* 
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^ TP TP TJ1 TP 

CO Cl H o 05 
Of Of Of Of J~f 

TP TPTPtPtP 

GC i> CO lO TP 

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Ty tT' ^y Tp 1 -ry 

ccoi^ooi 

7— < 7—» -t— < T— 1 O 

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CO t- GO 05 

CO CO CO CO {.— 

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CO i> GC 05 
i- i- L— i- GO 

Ol CO AO 

GO GC GC CXj GO 

CO i- 00 05 

GC' QC 00 00 05 


CO CO CO CO CO CO 

CO CO CO CO CO 

CO CO CO CO CO 

CO CO CO CO CO 

CO CO CO CO CO 

co co co co cd 

05 

.3 

O 00 CO CO ic CO 
lC y —« GO 1C Of 05 

i- © c> »n © 
CO CO — CC co 

Tt* c co co o 

CO rH GC co rr 

CO i- CO CO i> 

■»—< 05 c.— AO CO 

00 O Ol CO o 
7— o X CO AO 

34 

19 

04 

90 

77 

"5 Cd 
tf 

id co rr co oi © 

1- I- i'* i> i- i- 
rt rt rr rr rt 

05 GC £'• AO 

CO CO 50 ^ to 
'rf 1 rr rt 4 

CO ?' C C5 00 
CO CO CO iC IQ 
TP TP TT TP 

i- m co of 
in ic m m m 

TP TP Tp TP TP 

^OXNCO 
AO AO rf rt 1 ry 
'rf tt rr ^ rf 

in rt co T-I o 
TPrtTp Ttrp 
rt rt tt r* ^ 

6<U, 

© 01 ^ CO 00 O 

T-i 

Cl Tf O X o 

'<—* 1—t T— 1 T-H C? 

Of -rr CO GO O 
Of Of Of Of CO 

C) -P co X o 
CO CO CC CO Tf 

Ol '-t* CO GO O 
TP Tt Tp TP lO 

Ol rt CO 00 O 
m ao m in co 

« 

7“H 







f 


71 
































































































TABLE V.—RADII AND DEFLECTION ANGLES IN MINUTES PER FOOT OF LENGTH. 


t»o. 


o o o 

03 Tp 

O 

tH 
T* 


o o 
a* tp 


o o 

Cl TP 


o 

03 


o 

TJ» 


o o 

Cl Tp 


o o 

Cl T* 


o o 

C l rr 


o o 

Cl Tp 


o 

Cl 


o 

TT 


o o o 

Cl ZD 


03 

TP 


CO 

TP 


TP 

TP 


10 

TP 


co 

TP 




CO 

TP 


as 

TP 


o 

10 


«M Ut +* 

CO Tp lO CO i> GO 

as c t c? co 

Tp AO CO t- GO 

as o i-< ci co 

Tp aO CO GO 

as O y —' o co 

(£><1)0 

QP £ 

as as as as as as 

as o o o o 

o o o o o 

O T- 

P T—^ T-H 

' 1—1 T— < 

r—« T—< 

r-H Cl Cl ci 


i —• i—i i— < r-t 

T*H t—i i—i rH i—1 

T—I T—i 

t—i t— < i—1 

T—l T—> T—1 

*— * 1—1 

t— i r—> T—t r-i t—1 

w 

Tt n »o a w - 

tp o o co as 

GO —• l- *> 

o l> 

CO GC 03 

as an as o? go 

aO Tp CO O lO 


oo go as o o? tp 

co as o> ao ao 

Ci i> t- CO ^ 

i.'“ 03 CO TP i—■ 

!>• TP y— 

as co 

Tp ci co as i.'» 

a5 

T’ o? c as n m 

O0 i— 1 O GC CO 

ic co o? o as 

■*> co 

Tp CO O? 

o as go co io 

T CO 7) O Ci 

00 GO GO <. - 4 - i- 

t- SO co 

a co a co o 

aO lO 

AC »c AO 

AC T T T T 

Tt 1 TP Tp TP CO 

Ph 

y + < r—i rr ▼— i T" *H 

t—i l—< t—i r—i r—1 

1—1 1— 1 1—1 T—1 — ’ 

1— > 1“1 

T— 1 »- - 1— ' 

T— < 1—1 T— ' 

r " ^ 

T -l T— 1 T— < T— 1 T—1 


o o o o o 

o o o 

o o o 

o o 

o o 

O O 

o 

o o o o 

*U 

03 TP 03 TP 

o 

Cl TP 03 

TP 03 TP 

Cl TP 

Cl TP 

O? Tp 


TP C* TP CO 

Q 

!-• c? 

CO tP 

lO co 


N 

GO 

as 

o 

co co 

CO CO 

CO CO 


co 

CO 

CO 

TP 

u 

AO AO AO 

AO AO 

lO aO AO 

lO 

aO 

10 lO 

lO 

lO lO 

Q) 1) O 

go co as as o 

i— 1 t—« Cf Cl CO 

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CO O N i- X 

go as os 

o 

y—i y—t Cl Cl CO 

QP £ 

L- i- i> GO GO 

CO 00 GO GO GO 

00 GC OO GO GO 

CC GO GC 00 00 

go oooo as as 

as as as as as 

x 

p 

GO GO O T X T 

0? 03 CO SO T— 1 

i- tp co -p lb 

a: co Ci co -p 

-p 1C 

o — 

CMOTp-f 

'C 

<3 

co os a 7) x in 

ci as so co r—< 

OOOTWO 

CC 1- 

aO T CO. 

o ? r- o o as 

as GC 00 QC GO 

o 00 i— to TP CO 

o) o as cc i- 

iC T CO O) - 

a: oo so io 

■*T Of 0> 

— as 

oo < - CO iC rp 

— T-( — '-- 

’- o o o 

o o OO o 

Os Os 

as as as 

as as as as oo 

GC 00 QC 0C GO 

Ph 

a* ci ci ci ci ci 

0? Cl Cl Cl Cl 

O? 03 03 03 Cl 

1 —1 r—i 

i— * i— r—i 

»—< r— • l 

T—' T—< 

▼—« T—l T—< 


b o o o o o 

o o o o 

o o o o 

O O 

o o 

o o o 

o 

o o o o o 

bCrv 

® (s 

n^COTPlO 

O 

t— 03 CO TP 

io t— o* co 

TP AO 

i-i o? 

CO Tp AO 

»—1 

WCO'l’iGO 

o 

o 

t— 

co 


as 


o 


03 

03 

03 


03 


CO 




GO H ’tf i'* O CO 

co as ci ac go 

i—‘ Tp i> o co 

CO OS 03 io GO 

1—1 Tp O CO 

co as o ? io co 
• • • • 

(1) 0 O 

Q ^r 0 

• • • • • 

O* CO CO CO TP Tp 
Cl Cl Cl Cl Cl Cl 

TP Tp lC ic AO 
C? Cl Cl 

CO ZD CO i- t- 
O ? C l C l Cl Cl 

i— i — CO GO GO 
O? 03 03 03 0> 

as as a o o 

03 03 03 CC CO 

O O T-l —< —• 

CO CO co CO CO 

in 

P 

(^) y—> ZD 70 Cl Tp 
TP TP Tp aO SO l'- 

00 CO — 1— CO 
GO O Cl Tp CO 

qo i— co co as 

cc t-» co ci as 
03 co as co cc 

i— f'* GC O CO 
O TP GO CC l- 

X 7? C t- AC 
i— CO T“”i aC ^0 
• • 

■s 05 

si 

io tp co o> — o 
i> i> l- t- i> l- 

as as co i— co 
o co co so co 

io io tp co o? 
co co co co co 

03 i—■ o o cs 
CO CO CO CO aO 

as oo 1'- co 

AC aC AC AC lC 

CO AC lO Tp Tp 

AC aC iC aC O 

Deg. 

D 

o 

co n cc as o ^ 
J> i> L- 00 00 

« 

O? CO Tp AC co 
00 00 GO GO OD 

NGOCSOH 

oo co oc as as 

O? CO tP AO CO 
as as os as as 

j> go as o i-i 
as as as 

i—i i—i 

03 CO TP aC ZD 

o o o o o 

1—« »— 1—1 T—I 1—1 

«A-I t- -Pf ■ 

AO iO aO 

co rp co i - as o 

lO lO 

Cl CO AC SO 00 

io ia ao 
as ci tp io 

AO 1C 

I> GO O T CO 

AO O AC) 

tp co l- as o 

lC ao 

03 CO aO CO GO 

• • • 

D O 

QO GO GO 00 GC- as 

T—« r—« T —1 1—1 rH r—« 

as as as as as 

i—i i—i i—i i—i -rH 

as o o o o 

i-H Cl Cl Cl Cl 

>— 1 ■*—■ i—i 

03 03 03 O? 03 

i—< t— i-H t— < 0 3 

0> 03 03 03 03 

O? 03 O? 03 O? 

03 0? 03 03 03 

CC 

M' t>- GO lO -tp 

as r—. tp o as a 

CO -Tp lC 00 Cl 
AC 00 1—1 TT 00 

co o? as co ao 

I— AO GO O? co 

tPaCOGOO 
O TP CO Cl It 

tp go Tp as co 
i-i ac o tp as 

CO T-l o cs o 

TT as TP CC TP 
• • • • • 

•g 05 

M 

CO co o? ^ o o 
as as as as as as 

as oo oo *o> 

OO CO GO GO GO 

CD AC T T CO 
00 GO 00 GO 00 

CO O? i— i—• O 
GO GO GO GO 00 

o a as x n 

CC L— L- L- L— 

CO co AO ao 
i- £.- L- L- L- 

Q 

bo 

co 

30 

30 

30 

30 

30 

30 

30 

30 

30 

30 

30 

30 

o o o 
co co co 

l-H Cl 

co CO 

63 

64 

65 

66 

67 

68 

69 

70 

71 

03 CO 

l- i- 

74 

75 

«M *■« "g 

CO Tp AO CO 00 

OSOriOCO 

Tp aC CO t'- 00 

as o i— i o? co 

tp aC O It GO 

as o I-* o? co 

CD 0) O 

• *•••• 

1C 1C in AO in AO 

i—i 1—1 1—1 r—< i—1 1—1 

AC CO CO CO CO 

t— i i— i—i i—i r—• 

co co co co co 

t—i * —- i—i i—i i—i 

co i- 

1-H 1—1 T—1 1—1 1—1 

t- i- l- 

1—• 1—1 T—I T—■ 1—1 

L - GO 00 GO 00 

1—1 1—1 1—1 1—• T—1 

w 

p 

ic co o as as o 
co co as i—' tt oo 

i—i CO t— i— < co 

1— TP T— TP 

03 GO 1C CO Cl 
oo i— ao as co 

T-. c) CO T o 

N i~ aC O'. CC 

as co i- o? i- 

i— 03 CO 1— AO 

COON CO ^"P 

o ao as tp as 

■s 0 * 

M 

7^ - O O OS CO 
i—< i—< i— ( i— i o o 
1— ( 1—« 1—1 1— • »— 1—1 

GO i- CO CO aC 
O O O O O 

1—1 i—i y—< i—< i—1 

Tp TP co o? o> 
o o o o o 

1—1 1—1 T— > 1—1 »— 1 

i-i — o as as 
o o o as as 

1—1 T—I 1—1 

OC GO ? - CO 

as as as as as 

CO aC Tp -T co 
as as Os as as 


boo 

Cl TP 

o o 

TP 

o o o 

Cl Tp Ci 

o o o 

TP Cl TP 

o o 

03 TP 

OO 
03 M 1 

o o o 

03 tp 03 

o o o o 

Tp 03 Tp O 

ft 

T— < 

lO 

Cl 

AO 

53 

54 

55 

56 

57 

58 

59 

09 

CM ^ if 

CO Tp AC CO GO 

as o — ci co 

TP ao co GO 

os o t—i 03 co 

Tp lO CO It GO 

as o —- 03 co 

<X> <D O 

Cl Cl Cl Cl d Cl 

•»—( i—< t—< ri l—i l—< 

Cl co CO CO CO 
1—1 1—< 1—1 1—1 »—• 

CC CO CO CO CO 

1—1 T—I 1—1 T—I 1—1 

■CO T TP Tp —T' 

T— T—i 1—1 1—1 T—i 

Tp Tp —r Tp TP 
i—i i— i-H i-i i"T 

Tp *0 AO ao ao 

1-- 1—1 T— T—1 1—1 

c» 

P 

1C CO Cl CO AC O 
L- CO 1C TP CO CO 

lC CO Cl CO lC 
Cl Cl Cl Cl Cl 

# 

CO CO O It CO 
03 CO Tp T^I aO 

3 s - as o? »o 
co as o 03 

i- iC Tp if AO 

co ac i.'* as i—i 

N O Tp Oi AC 

CO CO 00 O CO 

■S“5 

« 

as co t- co ic -p 
co co co co co co 

TT 1—1 ■»—1 T-1 1—1 !—• 

CO O? i—* as 

CO CO CO CO o ? 
1—1 1—1 1—1 1—1 1—1 

GO N CC iC tP 
O? O? 03 Cl Cl 

i—i i— ( i—i i—i i—i 

CO O? 1- 1-H O 

O? ci 03 03 03 

1—1 1—1 1—1 1—1 1—1 

as oo co co 

i— y— tH — 1 i—< 

t—^ ^ r —1 

AC Tp CO CO 03 

1—> 1— 1 1—1 1—1 T— • 

1—1 1—1 T—1 1—1 1—1 


72 
























































































TABLE Y .-Continued. TABLE VT.—ARC EXCESS * TABLE VII. -TANGENTS AND EXTERNALS FOR A 1* CURVE. 


G 

CO 00 Oi o o co 

oocw-o 

00 CO t- Oi 1- o* 

?> Tt CO JO l> CO 
O Oi JO Oi rr i—• 
L^- Oi ir Oi Q0 rf 

CiCOiCCNN 
G3 00 QC C Oi TO 
C3 JO i— 00 rr © 

03 CO © 00 t> Oi 
- X C TP rr Q 
t- CO C i.- V n 

l'- JO CO 03 co co 

00 Oi i- CO 1 - o 

CC CO CO i— 03 

+-> 

w 

i- 00 00 03 03 © 

O i—» i—i Oi Oi co 

T“1 T—1 1—1 T—1 1—• 1—1 

co rr jo jc co 

1—1 1—1 T—1 1—1 1—• 1—1 

i- GO 03 05 o' — 
r—< r— i—i i—■ Oi Oi 

i— Oi co rh -<r jo 

Oi Oi Oi Oi Oi Oi 

4-3 






G 

<D 

C 

H 

C3 lO 1—i i>» CO 03 

wcocoi-o 

CO Oi Ci CO CO o 
Tt< GO r— JO C5 CO 

CO iO CO n C5 N 
CO O GO 1-1 O 

CO jo CO C> Oi 1-1 
03 CO L- 1— aO 03 

i—i — -.—i i— Oi Oi 

CO 1 .'- r- jC 03 CO 

© 00 J> JO CC Oi 
C O —■ C> CO Tf 
CO CO CO CO CO CO 

O GO l>- 1C TO Oi 
JO JO CO i- 0C 03 
CO CO CO VO co CO 

O 05 l- JO TT O) 
CO’-C^COrf 
TT 1 IT O Tf 

O 03 i>* CO TT Oi 
JO JO co t- QC 03 
rr rr rr rr rr rr 

t— 03 QC CC rf* CO 
© © i— Oi CO rr 

JC JC JO JO JC JO 

a* 

yxi 

© o o o o 

riCiCO^TO 

ooooo 
i-i Oi co rr jo 

ooooo 

r -1 Oi CO rr JO 

ooooo 

i—> Oi CO rr o 

ooooo 

T-1 Oi CO ^ lO 

< 

o 

CO 

1- 

GO 

03 

o 

1—1 


External 

E 

co CO CO © lO CO 
r- a X o o CO 
Oi Oi co rr co 

CO ’’t N Tf rH r-1 

i- Oi 00 CO JO JO 
OCOr-COiOt- 

co n it -f t>- 
CO GO 7> i- CC O 
03 1-1 rr CO 03 Oi 

C> O 03 O rr o 
o; 03 o: Oi jo o 

TP i.’ O V l- 1- 

CC 03 i—i JO CO CO 

JO Oi 1- O 1- CO 
rr GO Oi CO O rr 


1—1 1—1 1—1 T-H T—1 

1-1 Oi Oi Oi Oi CO 

CO CO rT rr rT JO 

JO JO CO CO J> 

4-3 






G 

0) 

G 

c3 

H 

O Tf N T- Tf CO 

o co © o co co 

Oi jo 03 co i> © 

© CO CO © CC L'- 

rr GO CO CO t—* JO 
O C7 t- C V N 

03 rr OC CO i> c> 

o rr j> i- rr oo 

J>COJ>COOOrr 

1- JO GO Oi JO 03 

oxoiocorn 
JO JO SO L- OQ 03 

OOCCOJOCO-^- 
© © t— Oi CO rr 

1—< 1—1 1—> 1—1 1—1 T—< 

O OC co JO CO !-• 
JO JO CO 1- QC 03 
1—• 1—1 1—' 1—1 1—1 1—1 

O GO CO JO CO i— 1 
CO'-OCOM' 

oi oi Oi oi Oi oi 

O GO CO JO CO 1-1 
lO JO CO 1- CC 03 

Oi Oi Oi Oi Oi Oi 

-2 

G < 

© © © © © 
^•CiCOTfiO 

o 

ooooo 
t— i oi co rr ao 

ooooo 

1-1 Oi CO TT JO 

ooooo 

i—i Oi CO iT JO 

ooooo 

1—1 Oi CO TT JO 

< 

TH 

Oi 

CO 

rT 

JO 


Actual 

Arc. 

One Sta. 

100.000 

100.001 

100.005 

100.011 

100.020 

100.032 

100.046 

100.062 

100.020 

100.026 

100.032 

100.038 
100.046 
100.054 
100.062 
loo. on 

100.020 

100 023 
100.026 
100.029 
100.032 

100.035 

100.038 

100 042 
100.046 
100.050 

100.054 

100.058 

100 062 
100.066 
100.071 

Deg. 

D 

o -rr Ci CO rT JO 

co i> oc 03 o 

i— Ci co rh jo 

CO t- QO 03 © 

1 — Oi o? rt JO 

CC t> GC C3 © 

1—1 

1—1 1—1 1—1 1—> 1—1 

1—1 T—< T-H 1—1 Oi 

Ci Oi Oi Oi Oi 

Oi Oi Oi Oi CO 


Def. 

per 

Foot. 

QOiH^NOCO 

oi Oi oi cd cd 

co CO CO CO CO co 

CO 03 Oi JO GO 

CO CO rT "T rr 
CO CO CO CO CO 

i— ' rT o CO 

JO JO JO TO CO 
CO CO CO CO CO 

CO C3 Oi JO GO 

COCOl-i-i- 
CO CO CO co co 

HTf NC CO 

CC QC 00 05 C3 
CC CO CO CO CO 

CO 03 Oi JO GO 

05 05000 

co co rr rr rr 

Radius 

R 

JO JO CO 03 Oi 

O JO O JO o to 

*** CO CO Oi 0> 1—i 
JO JO JO lO JO JO 

OrlOCOO 
h L— Oi GO 

▼H o O 03 03 
JO JO JO rr rr 

i>- CO JO JO CO 
03 jO t—i Ir CO 

00 00 00 l> 
rr rr rr rt rr 

N 03 r-1 rT N 
C3 JO Oi GO rr 

co co co jo jo 

r* r 1 r 1 rji 

Ci CO O* CO rr 
-i-rroi- 

JO rr rf rr CO 
rr rr rr rr rr 

1 — 03 CO rr CO 
rr O l- rr T- 

CO CC Oi Oi Oi 
rr rT rr rr rr 

fairs 

0) (s 

a 

o 

CONCOOOn 
O O O O h i—i 

i—1 i—i i—i tH i—i t—1 

Oi CO rr JO CO 

1—1 1—1 1—1 T—1 1—1 
1—1 1—1 1—1 TH 1—1 

GO 03 O -th 
HH rlO»W 
1—1 1—1 1—1 1—1 1—1 

Oi CO rr JO co 
oi oi Oi oi Ci 

H i—i rH i—i i—1 

GO 03 o 

Oi Oi Oi CO CO 

1—1 1—1 1—1 1—1 1—1 

Oi CO ir JO CO 

CO CO CO CO CO 

1—1 1—1 1—1 tH T—1 


73 


* See page 39. 





































TABLE VII.—TANGENTS AND EXTERNALS FOR A 1° CURVE. 


d 

OHO O CO Tf CO 
40 L- O 0^ »0 GO 

cci0 0:co05r- 

T-4 Tf i- rH tMX 

tO i> 05 0^ i- ^ 
O! tO O *C 05 rr 

C> t— i O? Tf OC CO 
05 Tt 05 C5 iO 

a GO GO 05 G-! to 
O tO O'! GO iO r-( 

r— < 05 H 05 G! 
GO Tf i— • GO iO CO 

53 

- 4-3 

w 

oc o co ic r- 05 

QC 05 05 05 05 05 
i—* rH t—* r—i i—i tH 

G! Tp CO 05 th CO 
OOOOt-h — 
0! G! G! G! G> G* 

tO GO r-4 CO iO GO 
rnnO C»WG' 
O! O! O! O! O! O! 

o co 1 C oc o co 

CO CO CO CO TF 

G! 0! O! O! O! O! 

tO GO’ CO tO 05 

r- it iO lO lO iO 
O! 0 > C! O! O! G ! 

1—1 Tt 1 t"- 05 G! iO 
tOOtOtCi^i-'- 
G! G! G! G! G! G! 

6s 

05 00 t- CO iO tP 

TP CO G! G! rH rH 

i-4 O O O O O 

1—4 t-4 1— ' i— • C^! G^ 

CO rt< iO tO J> 

XOHO?TtlO 

b£ 

, 

aS 

Eh 

^ O 05 GO l- tO 
GO 05 05 O — G! 

0*0 

r—4 i—4 r—i t—< i—i rH 

id tp co G! — o 

CO *f iO CC {» 00 
IO 40 40 iO iO 4C 

1—1 1-> T—1 r—1 1—1 1—4 

C5 GC ^ tO 40 

QC' 05 o — OJ co 
IO iO to to to to 

1—4 1—4 1—4 r-1 1—4 1—4 

CO 0! 1-4 O 05 oo 
40 tO i- GO 

to to to to to to 

1—1 1—4 1—4 1—4 1—4 1—4 

t- to id Tf CO G! 
o: O 1— G! CO TT 
tO i- i- t— i- i- 

T— 1—4 1—4 T—1 1—1 1—4 

ih i—i O 05 00 J>» 
iO to i>- GO 05 

i- i - J> i> l- C'- 

T-4 i—i 1—4 1—4 H IH 

<1 

a> 

© o o o o 

rs WCO 

o o o o o 

i— i <0! CO tp iO 

O O c G O 
-r-4 O! CO TT O 

• 

OOOOO 
1-1 0! CO ^ AO 

OOOOO 
1—4 G! CO Tji iO 

OOOOO 
T4 G! CO XJ4 iO 

fl 

o 

05 

G* 

30 

1—4 

CO 

32 

33 

34 

• 

e 

<X> 

05 CO GO TP G? th 

CO rH 00 50 Tp G! 

N05 0 0 ? Tp tO 
1H rs ^ O? OJ 

i—1 rH rH rH rH i - 4 

CO to O to ^ 
OXtO 40 CO o 

GO 05 •*—< CO iO iH 
C> O) co CO CO CO 

HHtHtH r-4 r-4 

O! O! 'tf tO O tO 
rsOC5XXH 

Ct — 7? d tO X 
O? rf -f Tf rr Tf 

1—( 1—4 1—4 1—4 T—• 1—4 

WOOHWt- 
L- 4.- i - 

o* oi Tfcoxo 

iO iO iO iQ iO to 

1—4 T—1 1—4 ' 1—4 1—4 

G’ GO tO tO tO GO 
GO 00 05 O th O! 

G! rf to 05 i-H CO 

to to to to i- 

T—4 T—4 1—1 T—4 1—4 1—4 

175.42 

177.56 

179.73 

181.90 

184.09 

186.30 

H 







6s 

00 AO h GO IO G! 

05 l> Tf r-s GO to 

CO i— x to CO i—* 

05 Tf O! O 00 

i> o co I—* o x 

tO iO Tt O? th O 

biD 

G 

c$ 

EH 

lCTfC0^0 05 
CO H GO O O O 
T-4 1—1 1—4 1—4 G! G! 

r—I r—l TH i—I i —4 r—4 

?> tO iO Tp G! i—i 
— G! CO Tp iO tO 
G* G! G! G? 0? G! 

1—4 1—4 1—4 i—4 1—4 1—4 

O C5 ^ to IO Tf4 

NHXC50-H 

C! O! 0! GN! CO CO 

1—1 1—4 1—4 1—4 1—4 1—1 

O! rH o 05 00 to 
O! CO Tf Tf iO tO 
CO CO CO CO CO co 

1—4 1—4 1—4 1—4 1—4 1—4 

iO ^ CO G! 1—4 05 
l'- QC 05 o r— 1 1—4 
CO CO CO rr ^ Tf 

1—1 1—4 r— T—t T—1 1—4 

00 i— tO i(G H' CO 
WCOrtiCtCi- 
rf rr -rr Tf rf Tt 

i—( i—i r—i 1—4 i—4 i—4 

<1 

<X> 

© o o o o 

r-4 G! CO TP iO 

© © © © © 
i—i CO! CO TP iO 

ooooo 

1-1 O! CO rf O 

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74 

















































TABLE VII.—TANGENTS AND EXTERNALS FOR A 1 ° CURVE. 



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75 















































TABLE VII.—TANGENTS AND EXTERNALS FOR A 1 ° CURVE. 


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76 








































TABLE VII.-TANGENTS AND EXTERNALS FOR A 1° CURVE. 


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CC CO C? rf iO N 

x -r o so 7) x 
i> a: — 7 ? Tf *r5 

7) 7) CO CO CO CO 
CO CO CO CO CO CO 

• 

r-uC C5 iO ’-I X 

lO—t-TP-rnt- 

t— 05 O 7) Tt< *0 

CO CO rf TT Tt TP 

CO CO CO CO CO co 

-4-3 

a 

© 

OOiOO^W 

C GO O O O O 


HN^fWOO 



— — « O O o C5 

C X X GO X X 

00 C5 05 05 o o 

O) CO Tt IQ 

ONXCidw 

o — co iC so x 

C7)’T^XO 

7) *T SC QC ▼—< CO 

lO l- C5 — CO lO 

i'- 05 r-« CO CO 00 

c3 

H 

lO CO SC CO SO so 

i- c- i- i- i> CO 

QO 'X' X X C5 05 

C5 C5 C5 o o o 

O O T—1 T-I T-H rH 

cckooidoco 

oooooo 

CD CO SO CO CO SO 

50 50 50 J><> i> 

i> i> i> i> fcr. i> 

© 

o o o o o 

o o o o o 

o o o o o 

o o o o o 

o o o o o 


th 7) CO Nr lO 

o 

t-h 7) CO ^ o 

T-iCiCO TT IO 

nWCOTj’O 

r^OCOTjiiO 


X 

C5 

o 

T—t 

7) 

05 

05 

o 

o 

o 




T-H 

rH 

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External 


7) O X *> i- t>- 

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C5 O t-. CO AO 

i7 5C SO SO 50 50 
0 0/0 0)0)0) 

X 05 rH rt t- O 

rnrtQo'rH-tQO 

t-xo5-oco 

OOOM--' 

O 0 ) 0 ) O) O) o 

IO O LO r-H x 50 
-r— io go o? id 05 

1C 50 i- 05 O tH 
t- t- i- X X 
7) 7) 7) 7) 7) 7) 

Tf (?) rH r-( O) CO 

co t-h io 05 co 
CO TP O t- X o 
X X X X X 05 
7) 7) 7) 7) 7) 01 

lONrH^C5^ 

i> T-H so O 05 

TH CO TP CO N X 

05 05 05 05 05 05 

7) 7) 7) 7) 7) 7) 

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to: 


0 ) x rf o/ o :o 

*>'50 Oi-XO 

O) io X O) *> O) 

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P*H 

X !C CO — 05 50 

Tf O? O X 50 IO 

CO r— 05 X 50 lO 

CO C» r- C5 co N 

50 io CC CO O) 


CO id - 05 O 0? 

50 x 05 — co 

lO *> X o o? 

50 x O r— co ir> 

t- 05 TH CO io N 

G 

H 


O O O O' --- 

50 50 50 50 50 50 

r- T-^ r— — O) O) 
SO 50 50 50 50 SO 

7> 7) 7) co co co 
so SO 50 50 50 50 

CO X ^ tp rr tJi 
so 50 50 50 50 50 

tP rt lO tO lO O 

SO 50 50 50 50 50 


© 


o o o o o 

o o o o o 

o O C3 o o 

o o o o o 

o o o o o 

]?« 

o 

co 

T-H 0) CO TT io 

nCICOTPO 

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n C/ CO ^ O 

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thOCOtPO nOCOTPiO 

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05 


05 

05 

05 

05 


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c 

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NCH>COON 

lOCOHOOO 

T-I O) Tt 50 05 CO 

50 T-H so O) X lO 


id 50 X 05 o 7 ) 

cd io 50 00 o ^ 

CO »C <*• C5 T-. CO 

id 05 HH co 50 

QC t-h co 50 00 t-h 

CO rf lO 50 X 05 

O — C) O? iC o 

X 05 O 7) SO 

•+10 50 00 0 0 

T-H CO TT io 50 X 

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7) 7? 7) 7) c? 7) 

CO CO CO CO CO CO 

CO CO .CO Tf T -r 

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7) 7) 7) 7) 7) 7) 

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7) 7) 7) 7) 7) 7) 

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TP io t- 05 7) TP 

05 CO i> O) X TP 


05 X 05 O 0) ^ 

50 05 0) 50 TH 50 

cd 05 io — x tP 

C l— CO O 50 CO 

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co o go m co 6 

CO -r SO X 05 r-H 

CO TP SC X 05 — 

CO O O QC 05 h. 

CO -P 50 X 05 H-. 

CO io 50 X O O 

G 

1C »C lO to O 50 

SO 50 SO 50 50 i'- 

1- L- t- ‘ - X 

X X X X X 05 

05 05 05 05 CO 

G 

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iC IO to O iC lO 

io »o *c O lO *o 

iO iC 1C io lO lO 

lO lO io O lO lO 

AO io io io 50 50 

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o o o o o 

o o o o o 

o o o o o 

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r-H 

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1-1 C) CO O iO 

T-OCO OlO 

t— i O) CO TF lO 

th O) CO lO 

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05 

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T-H 

C) 

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05 

05 

05 


p 

t'- lO ”TJT rf Tf Tf 

lO 50 X O 0 ) IO 

05 0 ) i> T-H so 0 ) 

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t-h o? O) co co rfi 

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05 o th o) cd 

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O) CO ’Tf *o 50 i — 

X 05 o T— O) co 

—T iO SO < H QO 05 

O TH 7 ) CO Tf IO 

50 X 05 O t-h O 

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05 05 05 05 05 05 

o; 05 o o o o 

o o o o o o 

TH T-H T—T T—I T-H t—H 

th t-h t-h 0 ) 7 ) 7 ) 

M 

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ri •— 1 t-h i-H t—» t —1 

TH T-H O) O) 0 ) C) 

O) 0 ) 7 ) O) O) O) 

7 ) 7 ) 7 ) 7 ) 7 ) 7 ) 

7 ) 7 ) 7 ) 7 ) 7 ) 7 ) 


4-3 

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m Tf co 7) 7? CO 

CO Tfr SO X O CO 

50 05 CO X 7) X 

CO 05 50 07 O X 

50 CO CO co CO 

© 

05 rt 05 05 ’t 

05 t* 05. Tpl O io 

o' lO — so 7) 

CO X ”+’ O 50* -rH 

N CO 05 »7 TH N 

50 X 05 TH 7') Hf 

lO l- X O 7> CO 

io SO X 05 — 7) 

7 m N 7 o w 

CO io 50 X O TH 

G 

O O O — T- t-h 

T-H T- TH O) 7) 7 ) 

7) 7) 7) 7) CO CO 

CO CO CO CO Tf 

Tf ’t T TT iO O 

G 

lO io U7 lO IO io 

O lO lO lO 1C IO 

lO IO io iO io io 

lomoiooio 

io io io io IO io 






© 

ooooo 

OOOOO 

ooooo 

ooooo 

ooooo 

1H 7) CO +P lO 

T-H (7) CO io 

T-H 7) X Nf AO 

T-H 7 ) CO 'PT 1 io 

T+ 7) CO +P IO 

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o 

CO 


io 

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X 

X 

X 

X 

X 

♦ 


77 


























































TABLE VII.—TANGENTS AND EXTERNALS FOR A 1° CURVE. 


<3 

a 

X 

W 


fl 

<X> 

a ^ 

<3 

H 


^^005^0 C0(XC5 0?C W OO^TfCOlO 


JO O? 05 CO Tf O? 
05 o? ti-oo? 
CO ”T ”T ~f *0 iO 
10 JO C O O iO 


05 L- jO Tf O* — 

jo r ^ -t i. o 

1C jO CO CO CO 

JO o o o o o 


OCJXi'^O 

CO JO GC-tl- 

t - i - i - CC GO GO 
iO JO JO iC aC o 


CO GO JO CO C? CO CON^TW^ t> 


l> Tf CO O 


CO l>- 05 • CO o 

OOOO^^Ci 

jO JO jc co co co 

05 05 05 05 05 05 


i- O OJ O h 
o> co 05 c? ocs 
1- I- 00 X GO 
05 05 05 05 05 05 


Tf GC — jO 05 CO 
O f i O 05 O ? lO 05 
05 05 05 O O O 
05 w* w. o o o 


1) 

If* 


o o o o o 
th o* co Tf jo 


o o o o o 

ri^CO^O 


o o o o o 

T— < <?l CO Tf jo 


o 

CO 


05 


o 

o? 


13 

a 

CO 1C lO CO 05 0? 

NTCnOOW 

Tf 05 Tf i—< 05 05 

OWOthoOO 

CO CO 05 C? GO lO 

K 

H 

r- 1 't i- O CO t- 
10 i- 05 0> Tf CO 

co co :oi-i-i- 

O rt- rx’ O) CO O 
OJ H CO c X ri 
i.'- CO GC 'GC GC 05 
V 1 t V ^ VT 

Tf QO CO GO O? 

CO »G GC O CO JO 
05 05 05 ‘-O CO 

V T TP JO O O 

CO GO CO 05 Tf o 
GO O CO iO GO — 

O TT 1—( * — • ’ — • O? 

iC jC JO 1C iC o 

CO O'* GC JO *— GO 

CO CO X 1— Tf CO 

O? O'? OJ CO CO CO 

JO O JJ5 iO JO O 


Tangent 

T 

TH Jr; O N JO Tf 

Tf 05 < - *C 

RO GO ^ CO 50 05 
CO 50 L- i- t- C- 
GC GO GO GC GC GC 

Tf CO 05 CO GO JO 

CO ■ 05 GC CO O 
Of Oi- O CO 50 
GC' GO GO 05 05 05 
GC GO GC GO GO GO 

CO CO CO JO GO CO 

-TOO-OO 
05 O? JO GO i— Tf 
C5 O O O ~ — 
GO C5 C5 O. 05 05 

05' {.- CO 50 GO rH 

05 05 05 05 05 

CO 05 O# JO GL 0? 
— —« 0» OJ 0? CO 
05 05 05 05 05 05 

IO TT 05 i> QO O 

O TT o? CO O 

jD GC ■— ~f <.— O 

CO CO Tf Tf Tf jQ 

05 05 05 05 05 05 





1 


<X> 

o o o o o 

O © © O O 

O O O O o 

o o o o o 

o o o o o 


r- 1 o? co Tf jo 

o 

tH 0 ? CO Tf iQ 

t—1 0? CO Tf *o 

i—i G? CO Tf iO 

T-H G? CO Tf O 


CO 

-f 

JO 

o 



tH 

T—1 

1—1 

T—1 

T 


rH 

TT 

T—1 

TT 

tT 

l"T 






c5 

a. 

jo o t> jo co co 

Tf AO GO O? CO CO 

O GC L- l- 00 h 

TfXiOWOC) 

C5 T- CO GC O? 05 

er 

£ 

GO GC i- i- i> 

l- i> GC GC 05 

o o — ci co jo 

CD N 05 th CO O' 

O 05 T CO CO GO 

*— CO JO 05 i— 

CO iO 05 — co 

O X O 5» O -O 

GO O C? id C5 

1— CO CO 00 o o* 

-i~> 

o o o o o — 

T—1 1-H — H 

0» o * CO co CO co 

CO Tf Tf -f Tf Tf 

lO JO JO JO co CO 

M 

H 

Tf Tf Tf Tf TT Tf 

Tf Tf Tf Tf Tf -p 

Tf Tf Tf Tf TT Tf 

Tf Tfi Tf Tf "T Tf 

Tf Tf Tf Tf Tf Tf 

4^> 






fl 

J> 05 h JO 05 iO 

O? 05 GO GO GO O 

CO ?> O? GO lO CO 

O? O? Tf CO O IO 

IT GO co CO CO GO 

0/ 






CO O *0 05 CO 00 

CO i- O* 1- c* GO 

CO 'GO "'"f '05 iC. t— 

i- CO 05 ito O? GO 

JO T x JO 0? 05 

GC »—> CO iQ GC O 

CO ‘O GO O 50 »o 

GO O CO O X —< 

CO CO 'GO T—■ Tf to 

05 o? -r i.- o o? 

Eh 

GO 05 05 05 05 CO 

© © o -< — 

— 0? o> c? o? CO 

CO CO CO ^ rr 

T jO lO iO o o 

t- L- i- c- L- GO 

CO GO GO GC GO GO 

GO GO GO GO GO GO 

GO GO GO GO GO 00 

00 GO 00 GO GO GO 

<X> 

© o o o o 

o © © © o 

O O O O O 

O O O O O 

o o o o o 


r-i C{ CO Tf o 

o 

T-< 0? CO Tf *0 

r-i O? CO Tf jQ 

H C/ CO o JG 

TT <0? CO Tf JO 


GO 

05 

O 

i—1 

c? 

<1 

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T-H 

th 

1—J 


T—1 

T—( 

T—1 

TT 

tT 


c3 

c 


CO JO Tf JO CO GO 

O Tf 05 Tf o GO 

CO Tf Tf lO CO 05 

O? CO O? GO Tf c? 

TH TT 0? Tf CO O 

It 


Tf TT GO *0 0? 05 

--- « 05 C- -n 

O? O GO CO Tf o? 

05 GO CO jo rf 

CO O? rT o 05 05 

0) 

U050^TO 

t- 05 —* O -r CO 

00 O — CO JO e- 

05 o Of —' CO GO 

O O? TT CO l— 05 

1 


Tf Tf lO JO jO iO 

jO O CO CO CO 50 

CO ?- i- l> c- i- 

< - GC X GO X GO 

05 05 05 05 05 05 

M 

H 


CO CO co CO 50 CO 

CO 70 CO CO co CO 

co co co co co co 

CO CO CO CO co 70 

CO CO CO CO 70 co 


Tangent 

T 

CO O? X JO CO h 

CO iO CO 00 CD O* 
O o “T 50 05 — 
O? O? D? 0> C> CO 
i.- c- i> i_- i- c- 

H H c? f U O 

•POXO *''0 
CO JO i - CD C? -f 
CO CO CO T TP T 
l- t- i - L'- c- c- 

jC O CO' CO tt o 

i - o> JO CC -T 

CO 05 — CO JO 00 
*r jd iO jC O 

L- L- c- l- V- t- 

OOWf NOi 
Tf i'- r^, rif O 

O C? »c i- 05 Oi 

CO CO CO CO CO 
i- i— i- L- C- X- 

ocooxoo 

CO tt Tf GC 1 O? 

Tf CO 05 — 70 CO 

GO 00 CO 

L- L- L- c- L- 

0) 

o o o o o 

O O O O O 

o o o o o 

o o o o o 

o o o o o 


it 0? CO Tf jO 

o 

i—i o? co Tf jo 

T-H o? CO ^ JO 

T-t C? CO Tf JO 

i—i C? CO Tf iO 


cO 

Tf 

JO 

co 


<5 

o 

o 

o 

o 

o 

# 

T — 1 

tT 

T 

tT 

TT 


• 


78 




% 

























































TABLE VIII . REDUCTION OF THE MERCURIAL BAROMETER 
TO STANDARD FOR TEMPERATURE . 


Att . 

Height of the barometer in inches . 

Ther . 

19 

20 

21 

22 










23 

24 

25 

26 

27 

28 

29 

30 

31 

F . 

In . 

In . 

In . 

In . 

In . 

In . 

In . 

In . 

• In 

In 

In 

In 

In . 

0 ° 

+ .050 

.052 

.055 

.057 

.060 

.063 

.065 

.068 

.070 

.073 

.076 

.078 

.081 

2 

.046 

.049 

.051 

.053 

.056 

.058 

.061 

.063 

.065 

.068 

.070 

.073 

.075 

4 

.043 

.045 

.047 

.049 

.052 

.054 

.056 

.058 

.061 

.063 

.065 

.067 

.070 

6 

.039 

.041 

.043 

.045 

.047 

.049 

.052 

.054 

.056 

.058 

.060 

.062 

.064 

8 

.036 

.038 

.039 

.041 

.043 

.045 

.047 

.049 

.051 

.053 

.054 

.056 

.058 

10 

.032 

.034 

.036 

.037 

.039 

.041 

.042 

.044 

.046 

.047 

.049 

.051 

.053 

12 

+ .029 

.030 

.032 

.033 

.035 

.036 

.038 

.039 

.041 

.042 

.044 

.045 

.047 

14 

.025 

.027 

.028 

.029 

.031 

.032 

.033 

.035 

.036 

.037 

.039 

.040 

.041 

16 

.022 

.023 

.024 

.025 

.026 

.028 

.029 

.030 

.031 

.032 

.033 

.034 

.036 

18 

.018 

.019 

.020 

.021 

.022 

.023 

.024 

.025 

.026 

.027 

.028 

.029 

.030 

20 

.015 

.016 

.016 

.017 

.018 

.019 

.020 

.020 

.021 

.022 

.023 

.024 

.024 

22 

+ .011 

.012 

.013 

.013 

.014 

.014 

.015 

.016 

.016 

.017 

.017 

.018 

.019 

24 

.008 

.008 

.009 

.009 

.010 

.010 

.011 

Oil 

Oil 

.012 

.012 

.013 

.013 

26 

.005 

.005 

.005 

.005 

.005 

.006 

.006 

.006 

.006 

.007 

.007 

.007 

.007 

28 

+ .001 

.001 

.001 

.001 

.001 

.001 

.001 

.001 

.002 

.002 

.002 

.002 

.002 

30 

-.002 

.002 

.003 

.003 

.003 

.003 

.003 

.003 

.003 

.003 

.004 

.004 

.004 

32 

-.006 

.006 

.006 

.007 

.007 

.007 

.008 

.008 

.008 

.009 

.009 

.009 

.009 

34 

.009 

.010 

.010 

.011 

.011 

.012 

.012 

.013 

.013 

.014 

.014 

.015 

.015 

36 

.013 

.013 

.014 

.015 

.015 

.016 

.017 

.017 

.018 

.019 

.019 

.020 

.021 

38 

.016 

.017 

.018 

.019 

.020 

.020 

.021 

.022 

.023 

.024 

.025 

.026 

.026 

40 

.020 

.021 

.022 

.023 

.024 

.025 

.026 

.027 

.028 

.029 

.030 

.031 

.032 

42 

-.023 

.024 

.025 

.027 

.028 

.029 

.030 

.032 

.033 

.034 

.035 

.036 

.038 

44 

.026 

.028 

.029 

.031 

.032 

.033 

.035 

.036 

.038 

.039 

.040 

.042 

.043 

46 

.030 

.031 

.033 

.035 

.036 

.038 

.039 

.041 

.043 

.044 

.046 

.047 

.049 

48 

.033 

.035 

.037 

.039 

.040 

.042 

.044 

.046 

.047 

.049 

.051 

.053 

.054 

50 

.037 

.039 

.041 

.043 

.045 

.046 

.048 

.050 

.052 

.054 

.056 

.058 

.060 

52 

-.040 

.042 

.044 

.047 

.049 

.051 

.053 

.055 

.057 

.059 

.061 

.064 

.066 

54 

.044 

.046 

.048 

.051 

.053 

.055 

.057 

.060 

.062 

.064 

.067 

.069 

.071 

56 

.047 

.050 

.052 

.055 

.057 

.060 

.062 

.064 

.067 

.069 

.072 

.074 

.077 

58 

.051 

.053 

.056 

.059 

.061 

.064 

.066 

.069 

.072 

.074 

.077 

.080 

.082 

60 

.054 

.057 

.060 

.062 

.065 

.068 

.071 

.074 

.077 

.080 

.082 

.085 

.088 

62 

-.057 

.060 

.063 

.066 

.069 

.073 

.076 

.079 

.082 

.085 

.088 

.091 

.094 

64 

.061 

.064 

.067 

.070 

.074 

.077 

.080 

.083 

.086 

.090 

.093 

.096 

.099 

66 

.064 

.068 

.071 

.074 

.078 

.081 

.085 

.088 

.091 

.095 

.098 

.101 

.105 

68 

.068 

.071 

.075 

.078 

.082 

.085 

.089 

.093 

.096 

.100 

.103 

.107 

.110 

70 

.071 

.075 

.079 

.082 

.086 

.090 

.094 

.097 

.101 

.105 

.109 

.112 

.116 

72 

-.075 

.078 

.082 

.086 

.090 

.094 

.098 

.102 

.106 

.110 

.114 

.118 

.122 

74 

.078 

.082 

.086 

.090 

.094 

.098 

.103 

.107 

.111 

.115 

.119 

.123 

.127 

76 

.081 

.086 

.090 

.094 

.098 

.103 

.107 

.111 

.116 

.120 

.124 

.128 

.133 

78 

.085 

.089 

.094 

.098 

.103 

.107 

.112 

.116 

.120 

.125 

.129 

.134 

.138 

80 

.088 

.093 

.097 

.102 

.107 

.111 

.116 

.121 

.125 

.130 

.135 

.139 

.144 

82 

-.092 

.096 

.101 

.108 

.111 

.116 

.121 

.125 

.130 

.135 

.140 

.145 

.149 

84 

.095 

.100 

.105 

.110 

.115 

.120 

.125 

.130 

.135 

.140 

.145 

.150 

.155 

86 

.098 

.104 

.109 

.114 

.119 

.124 

.130 

.135 

.140 

.145 

.150 

.155 

.161 

88 - 

.102 

.107 

.113 

.118 

.123 

.129 

.134 

.139 

.145 

.150 

.155 

.161 

.166 

90 

.105 

.111 

.116 

.122 

.127 

.133 

.138 

.144 

.150 

.155 

.161 

.166 

.172 

92 

- .109 

.114 

.120 

.126 

.132 

.137 

.143 

.149 

.154 

.160 

166 

.172 

.177 

94 

.112 

.118 

.124 

.130 

.136 

.142 

.147 

.153 

.159 

.165 

171 

.177 

.183 

96 

.115 

.122 

.128 

.134 

.140 

.146 

.152 

.158 

.164 

170 

176 

182 

188 

98 

.119 

.125 

.131 

.138 

.144 

.150 

.156 

.163 

.169 

175 

181 

188 

194 

100 

.122 

.129 

.135 

.142 

.148 

.154 

.161 

.167 

.174 

180 

187 

193 

200 


79 


























































TABLE IX. BAROMETRIC LEVELING. 
Values of 62583.6 (log 29.90-log B). 


B 

.00 

.01 

.02 

.03 

.04 

.05 

.06 

.07 

.08 

.09 

Inches. 

Feet. 

Feet. 

Feet. 

F eet.. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

17.00 

15347 

15331 

15315 

15299 

15283 

15267 

15251 

15235 

15219 

15203 

17.10 

15187 

15172 

15156 

15140 

15124 

15108 

15092 

15076 

15061 

15045 

17.20 

15029 

15013 

14997 

14982 

14966 

14950 

14934 

14919 

14903 

14887 

17.30 

14871 

14856 

14840 

14824 

14809 

14793 

14777 

14762 

14746 

14730 

17.40 

14715 

14699 

14684 

14668 

14652 

14637 

14621 

14606 

14590 

14575 

17.50 

14559 

14544 

14528 

14512 

14497 

14481 

14466 

14451 

14435 

14420 

17.60 

14404 

14389 

14373 

14358 

14342 

14327 

14312 

14296 

14281 

14266 

17.70 

14250 

14235 

14219 

14204 

14189 

14173 

14158 

14143 

14128 

14112 

17.80 

14097 

14082 

14067 

14051 

14036 

14021 

14006 

13990 

13975 

13960 

17.90 

13945 

13930 

13914 

13899 

13884 

13869 

13854 

13839 

13824 

13808 

18.00 

13793 

13778 

13763 

13748 

13733 

13718 

13703 

13688 

13673 

13658 

18.10 

13643 

13628 

13613 

13598 

13583 

13568 

13553 

13538 

13523 

13508 

18.20 

13493 

13478 

13463 

13448 

13433 

13418 

13404 

13389 

13374 

13359 

18.30 

13344 

13329 

13314 

13300 

13285 

13270 

13255 

13240 

12226 

13211 

18.40 

13196 

13181 

13166 

13152 

13137 

13122 

13107 

13093 

13078 

13063 

18.50 

13049 

13034 

13019 

13005 

12990 

12975 

12961 

12946 

12931 

12917 

18.60 

12902 

12888 

12873 

12858 

12844 

12829 

12815 

12800 

12785 

12771 

18.70 

12756 

12742 

12727 

12713 

12698 

12684 

12669 

12655 

12640 

12626 

18.80 

12611 

12597 

12583 

12568 

12554 

12539 

12525 

12510 

12496 

12482 

18.90 

12467 

12453 

12438 

12424 

12410 

12395 

12381 

12367 

12352 

12338 

19.00 

12324 

12310 

12295 

12281 

12267 

12252 

12238 

12224 

12210 

12195 

19.10 

12181 

12167 

12153 

12138 

12124 

12110 

12096 

12082 

12068 

12053 

19.20 

12039 

12025 

12011 

11997 

11983 

11969 

11954 

11940 

11926 

11912 

19.30 

11898 

11884 

11870 

11856 

11842 

11828 

11814 

11800 

11786 

11772 

19.40 

11758 

11744 

11730 

11716 

11702 

11688 

11674 

11660 

11646 

11632 

19.50 

11618 

11604 

11590 

11576 

11562 

11548 

11534 

11520 

11507 

11493 

19.60 

11479 

11465 

11451 

11437 

11423 

11410 

11396 

11382 

11368 

11354 

19.70 

11340 

11327 

11313 

11299 

11285 

11272 

11258 

11244 

11230 

11217 

19.80 

11203 

11189 

11175 

11162 

11148 

11134 

11121 

11107 

11093 

11080 

19.90 

11066 

11052 

11039 

11025 

11011 

10998 

10984 

10970 

10957 

10943 

20.00 

10930 

10916 

10903 

10889 

10875 

10862 

10848 

10835 

10821 

10808 

20.10 

10794 

10781 

10767 

10754 

10740 

10727 

10713 

10700 

10686 

10673 

20.20 

10659 

10646 

10632 

10619 

10605 

10592 

10579 

10565 

10552 

10538 

20.30 

10525 

10512 

10498 

10485 

10472 

10458 

10445 

10431 

10418 

10405 

20.40 

10391 

10378 

10365 

10352 

10338 

10325 

10312 

10298 

10285 

10272 

20.50 

10259 

10245 

10232 

10219 

10206 

10192 

10179 

10166 

10153 

10139 

20.60 

10126 

10113 

10100 

10087 

10074 

10060 

10047 

10034 

10021 

10008 

20.70 

9995 

9982 

9968 

9955 

9942 

9929 

9916 

9903 

9890 

9877 

20.80 

9864 

9851 

9838 

9825 

9812 

9799 

9786 

9772 

9759 

9746 

20.90 

9733 

9720 

9707 

9694 

9681 

9668 

9655 

9642 

9629 

9617 

21.00 

9604 

9591 

9578 

9565 

9552 

9539 

9526 

9513 

9500 

9487 

21.10 

9474 

9462 

9449 

9436 

9423 

9410 

9397 

9384 

9372 

9359 

21.20 

9346 

9333 

9320 

9307 

9295 

9282 

9269 

9256 

9244 

9231 

21.30 

9218 

9205 

9193 

9180 

9167 

9154 

9142 

9129 

9116 

9103 

21.40 

9091 

9078 

9065 

9053 

9040 

9027 

9015 

9002 

8989 

8977 

21.50 

8964 

8951 

8939 

8926 

8913 

8901 

8888 

8876 

8863 

8850 

21.60 

8838 

8825 

8813 

8800 

8788 

8775 

8762 

8750 

8737 

8725 

21.70 

8712 

8700 

8687 

8675 

8662 

8650 

8637 

8625 

8612 

8600 

21.80 

8587 

8575 

8562 

8550 

8538 

8525 

8513 

8500 

8488 

8475 

21.90 

8463 

8451 

8438 

8426 

8413 

8401 

8389 

8376 

8364 

8352 

22.00 

8339 

8327 

8314 

8302 

8290 

8277 

8265 

8253 

8240 

8228 


80 































TABLE IX. BAROMETRIC LEVELING (Continued). 
Values of 62583.6 (log 29.90 — log B ). 


B 

.00 

.01 

.02 

.03 

.04 

.05 

.06 

.07 

OC 

o 

.09 

Inches. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

F eet. 

Feet. 

Feet. 

Feet 

22.00 

8339 

8327 

8314 

8302 

8290 

8277 

8265 

8253 

8240 

8228 

22.10 

8216 

8204 

8191 

8179 

8167 

8154 

8142 

8130 

8118 

8105 

22.20 

8093 

8081 

8069 

8056 

8044 

8032 

8020 

8008 

7995 

7983 

22.30 

7971 

7959 

7947 

7935 

7922 

7910 

7898 

7886 

7874 

7862 

22.40 

7849 

7837 

7825 

7813 

7801 

7789 

7777 

7765 

7753 

7740 

22.50 

7728 

7716 

7704 

7692 

7680 

7668 

7656 

7644 

7632 

7620 

22.60 

7608 

7596 

7584 

7572 

7560 

7548 

7536 

7524 

7512 

7500 

22.70 

7488 

7476 

7464 

7452 

7440 

7428 

7416 

7404 

7392 

7380 

22.80 

7368 

7356 

7345 

7333 

7321 

7309 

7297 

7285 

7273 

7261 

22.90 

7249 

7238 

7226 

7214 

7202 

7190 

7178 

7166 

7155 

7143 

23.00 

7131 

7119 

7107 

7096 

7084 

7072 

7060 

7048 

7037 

7025 

23.10 

7013 

7001 

6990 

6978 

6966 

6954 

6943 

6931 

6919 

6907 

23.20 

6896 

6884 

6872 

6861 

6849 

6837 

6825 

6814 

6802 

6790 

23.30 

6779 

6767 

6755 

6744 

6732 

6721 

6709 

6697 

6686 

6674 

23.40 

6662 

6651 

6639 

6628 

6616 

6604 

6593 

6581 

6570 

6558 

23.50 

6546 

6535 

6523 

6512 

6500 

6489 

6477 

6466 

6454 

6443 

23.60 

6431 

6420 

6408 

6397 

6385 

6374 

6362 

6351 

6339 

6328 

23.70 

6316 

6305 

6293 

6282 

6270 

6259 

6247 

6236 

6225 

6213 

23.80 

6202 

6190 

6179 

6167 

6156 

6145 

6133 

6122 

6110 

6099 

23.90 

6088 

6076 

6065 

6054 

6042 

6031 

6020 

6008 

5997 

5986 

24.00 

5974 

5963 

5952 

5940 

5929 

5918 

5906 

5895 

5884 

5872 

24.10 

5861 

5850 

5839 

5827 

5816 

5805 

5794 

5782 

5771 

5760 , 

24.20 

5749 

5737 

5726 

5715 

5704 

5693 

5681 

5670 

5659 

5648 

24.30 

5637 

5625 

5614 

5603 

5592 

5581 

5570 

5558 

5547 

5536 

24.40 

5525 

5514 

5503 

5492 

5480 

5469 

5458 

5447 

5436 

5425 

24.50 

5414 

5403 

5392 

5381 

5369 

5358 

5347 

5336 

5325 

5314 

24.60 

5303 

5292 

5281 

5270 

5259 

5248 

5237 

5226 

5215 

5204 

24.70 

5193 

5182 

5171 

5160 

5149 

5138 

5127 

5116 

5105 

5094 

24.80 

5083 

5072 

5061 

5050 

5039 

5028 

5017 

5006 

4995 

4985 

24.90 

4974 

4963 

4952 

4941 

4930 

4919 

4908 

4897 

4886 

4876 

25.00 

4865 

4854 

4843 

4832 

4821 

4810 

4800 

4789 

4778 

4767 

25.10 

4756 

4745 

4735 

4724 

4713 

4702 

4691 

4681 

4670 

4659 

25.20 

4648 

4637 

4627 

4616 

4605 

4594 

4584 

4573 

4562 

4551 

25.30 

4540 

4530 

4519 

4508 

4498 

4487 

4476 

4465 

4455 

4444 

25.40 

4433 

4423 

4412 

4401 

4391 

4380 

4369 

4358 

4348 

4337 

25.50 

4326 

4316 

4305 

4295 

4284 

4273 

4263 

4252 

4241 

4231 

25.60 

4220 

4209 

4199 

4188 

4178 

4167 

4156 

4146 

4135 

4125 

25.70 

4114 

4104 

4093 

4082 

4072 

4061 

4051 

4040 

4030 

4019 

25.80 

4009 

3998 

3988 

3977 

3966 

3956 

3945 

3935 

3924 

3914 

25.90 

3903 

3893 

3882 

3872 

3861 

3851 

3841 

3830 

3820 

3809 

26.00 

3799 

3788 

3778 

3767 

3757 

3746 

3736 

3726 

3715 

3705 

26.10 

3694 

3684 

3674 

3663 

3653 

3642 

3632 

3622 

3611 

3601 

26.20 

3590 

3580 

3570 

3559 

3549 

3539 

3528 

3518 

3508 

3497 

26.30 

3487 

3477 

3466 

3456 

3446 

3435 

3425 

3415 

3404 

3394 

26.40 

3384 

3373 

3363 

3353 

3343 

3332 

3322 

3312 

3301 

3291 

26.50 

3281 

3270 

3260 

1 

3250 

3240 

3230 

3219 

3209 

3199 

3189 


81 











































TABLE IX. BAROMETRIC LEVELING (Continued). 
Values of 62583.6 (log 29.90^- IogR). 


B 

.00 

.01 

.02 

.03 

.04 

.05 

.06 

.07 

.08 

.09 

Inches 

Feet. 

F eet 

Feet 

Feet 

Feet 

Feet 

Feet 

Feet 

Feet 

. Feet. 

26.50 

3281 

327C 

3260 

325C 

324C 

323C 

321£ 

320S 

3196 

3189 

26.60 

3179 

3168 

3158 

3148 

3138 

3128 

31L 

3107 

309' 

3087 

26.70 

3077 

3066 

3056 

3046 

3036 

3026 

3016 

3005 

2995 

2985 

26.80 

2975 

2965 

2955 

2945 

293^ 

2924 

29L 

2904 

2891 

2884 

26.90 

2874 

2864 

2854 

2843 

2833 

2823 

2813 

2803 

279S 

2783 

27.00 

2773 

2763 

2753 

2743 

2733 

2723 

2713 

2703 

2692 

2682 

27.10 

2672 

2662 

2652 

2642 

2632 

2622 

2612 

2602 

2592 

2582 

27.20 

2572 

2562 

2552 

2542 

2532 

2522 

2512 

2502 

2493 

2483 

27.30 

2473 

2463 

2453 

2443 

2433 

2423 

2413 

2403 

2393 

2383 

27.40 

2373 

2363 

2353 

2343 

2334 

2324 

2314 

2304 

2294 

2284 

27.50 

2274 

2264 

2254 

2245 

2235 

2225 

2215 

2205 

2195 

2185 

27.60 

2176 

2166 

2156 

2146 

2136 

2126 

2116 

2107 

2097 

2087 

27.70 

2077 

2067 

2058 

2048 

2038 

2028 

2018 

2009 

1999 

1989 

27.80 

1979 

1970 

1960 

1950 

1940 

1930 

1921 

1911 

1901 

1891 

27.90 

1882 

1872 

1862 

1852 

1843 

1833 

1823 

1814 

1804 

1794 

28.00 

1784 

1775 

1765 

1755 

1746 

1736 

1726 

1717 

1707 

1697 

28.10 

1688 

1678 

1668 

1859 

1649 

1639 

1630 

1620 

1610 

1601 

28.20 

1591 

1581 

1572 

1562 

1552 

1543 

1533 

1524 

1514 

1503 

28.30 

1495 

1485 

1476 

1466 

1456 

1447 

1437 

1428 

1418 

1404 

28.40 

1399 

1389 

1380 

1370 

1361 

1351 

1342 

1332 

1322 

1318 

28.50 

1303 

1294 

1284 

1275 

1265 

1256 

1246 

1237 

1227 

1218 

28.60 

1208 

1199 

1189 

1180 

1170 

1161 

1151 

1142 

1132 

1123 

28.70 

1113 

1104 

1094 

1085 

1075 

1066 

1057 

1047 

1038 

1028 

28.80 

1019 

1009 

1000 

990 

981 

972 

962 

953 

943 

934 

28.90 

925 

915 

906 

896 

887 

878 

868 

859 

849 

840 

29 00 

831 

821 

812 

803 

793 

784 

775 

765 

756 

746 

29.10 

737 

728 

718 

709 

700 

690 

681 

672 

663 

653 

29.20 

644 

635 

625 

616 

607 

597 

588 

579 

570 

560 

29.30 

■ 551 

542 

532 

523 

514 

505 

495 

486 

477 

468 

29.40 

458 

449 

440 

431 

421 

412 

403 

394 

384 

375 

29.50 

366 

357 

348 

338 

329 

320 

311 

3021 

292 

283 

29.60 

274 

265 

256 

247 

237 

228 

219 

210 

201 

192 

29.70 

182 

173 

164 

155 

146 

137 

128 

118 

109 

100 

29.80 

+ 91 

+ 82 

+ 73 

+ 64 

+ 55 

+ 45 

+ 36 

+ 27 

+ 18 

+ 9 

29.90 

0 

- 9 

- 18 

- 27 

- 36 

- 45 

- 55 

- 64 

- 73 

- 82 

30.00 

- 91 

-100 

-109 

- 118 

- 127 

- 136 

- 145 

- 154 

- 163 

- 172 

30.10 

-181 

-190 

-199 

- 208 

- 217 

- 226 

- 235 

- 244 

- 253 

- 262 

30.20 

-271 

-280 

-289 

- 298 

- 307 

- 316 

- 325 

- 334 

- 343 

- 352 

30.30 

-361 

-370 

-379 

- 388 

- 397 

- 406 

- 415 

- 424 

- 433 

- 442 

30.40 

-451 

-460 

-469 

- 478 

- 486 

- 495 

- 504 

- 513 

- 522 

- 531 

30.50 

-540 

-549 

-558 

- 567 

- 576 

- 585 

- 593 

- 602 

- 611 

- 620 

30.60 

-629 

-638 

-647 

- 656 

- 665 

- 673 

- 682 

- 691 

- 700 

- 709 

30.70 

-718 

-727 

-735 

- 744 

- 753 

- 762 

- 771 

- 780 

- 788 

- 797 

30.80 

-806 

-815 

-824 

- 833 

- 841 

- 850 

- 859 

- 868 

- 877 

- 885 

30.90 

-894 

-903 

-912 

- 921 

- 929 

- 938 

- 947 

- 956 

- 964 

- 973 

31.00 

-982 

-991 

-999 

-1008 

— 1017 

-1026 

-1035 

-1043 

-1052 

-1061 


82 













































TABLE X. BAROMETRIC LEVELING. 
Correction for Temperature: + above 50° F. — below 50° F. 



Approximate Difference of Height from Table IX. 



20 

40 

60 

80 

100 

200 

300 

400 

500 

600 

700 

800 

900 

F. 

F. 

Ft. 

Ft. 

Ft. 

Ft. 

Ft. 

Ft. 

Ft. 

Ft. 

Ft 

Ft. 

Ft 

Ft. 

Ft. 

49° 

51° 







1 

1 

1 

1 

1 

2 

2 

48 

52 






1 

1 

2 

2 

2 

3 

3 

4 

47 

53 





1 

1 

2 

2 

3 

4 

4 

5 

6 

46 

54 




1 

1 

2 

2 

3 

4 

5 

6 

7 

7 

45 

55 



1 

1 

1 

2 

3 

4 

5 

6 

7 

8 

9 

44 

56 



1 

1 

1 

2 

4 

5 

6 

7 

9 

10 

11 

43 

57 


1 

1 

1 

1 

3 

4 

6 

7 

9 

10 

11 

13 

42 

58 


1 

1 

1 

2 

3 

5 

7 

8 

10 

11 

13 

15 

41 

59 


1 

1 

1 

2 

4 

6 

7 

9 

11 

13 

15 

17 

40 

60 


1 

1 

2 

2 

4 

6 

8 

10 

12 

14 

16 

18 

39 

61 


1 

1 

2 

2 

4 

7 

9 

11 

13 

16 

18 

20 

38 

62 


1 

1 

2 

2 

5 

7 

10 

12 

15 

17 

20 

22 

37 

63 

1 

1 

2 

2 

3 

5 

8 

11 

13 

16 

19 

21 

24 

36 

64 

1 

1 

2 

2 

3 

6 

9 

11 

14 

17 

20 

23 

26 

35 

65 

1 

1 

2 

2 

3 

6 

9 

12 

15 

18 

21 

24 

28 

34 

66 

1 

1 

2 

3 

3 

7 

10 

13 

16 

20 

23 

26 

29 

33 

67 

1 

1 

2 

3 

3 

7 

10 

14 

17 

21 

24 

28 

31 

32 

68 

1 

1 

2 

3 

4 

7 

11 

15 

18 

22 

26 

29 

33 

31 

69 

1 

2 

2 

3 

4 

8 

12 

15 

19 

23 

27 

31 

35 

30 

70 

1 

2 

2 

3 

4 

8 

12 

16 

20 

24 

29 

33 

37 

29 

71 

1 

2 

3 

3 

4 

9 

13 

17 

21 

26 

30 

34 

39 

28 

72 

1 

2’ 

3 

4 

4 

9 

13 

18 

22 

27 

31 

36 

40 

27 

73 

1 

2 

3 

4 

5 

9 

14 

19 

23 

28 

33 

38 

42 

26 

74 

1 

2 

3 

4 

5 

10 

15 

20 

24 

29 

34 

39 

44 

25 

75 

1 

2 

3 

4 

5 

10 

15 

20 

25 

31 

36 

41 

46 

24 

76 

1 

2 

3 

4 

5 

11 

16 

21 

27 

32 

37 

42 

48 

23 

77 

1 

2 

3 

4 

6 

11 

17 

22 

28 

33 

39 

44 

50 

22 

78 

1 

2 

3 

5 

6 

11 

17 

23 

29 

34 

40 

46 

51 

21 

79 

1 

2 

4 

5 

6 

12 

18 

24 

30 

35 

41 

47 

53 

20 

80 

1 

2 

4 

5 

6 

12 

18 

24 

31 

37 

43 

49 

55 

19 

81 

1 

3 

4 

5 

6 

13 

19 

25 

32 

38 

44 

51 

57 

18 

82 

1 

3 

4 

5 

7 

13 

20 

26 

33 

39 

46 

52 

59 

17 

83 

1 

3 

4 

5 

7 

13 

20 

27 

34 

40 

47 

54 

61 

16 

84 

1 

3 

4 

6 

7 

14 

21 

28 

35 

42 

49 

55 

62 

15 

85 

1 

3 

4 

6 

7 

14 

21 

29 

36 

43 

50 

57 

64 

14 

86 

1 

3 

4 

6 

7 

15 

22 

29 

37 

44 

51 

59 

66 

13 

87 

2 

3 

5 

6 

8 

15 

23 

30 

38 

45 

53 

60 

68 

12 

88 

2 

3 

5 

6 

8 

15 

23 

31 

39 

46 

54 

62 

70 

11 

89 

2 

3 

5 

6 

8 

16 

24 

32 

40 

48 

56 

64 

72 

10 

90 

2 

3 

5 

7 

8 

16 

24 

33 

41 

49 

57 

65 

73 

9 

91 

2 

3 

5 

7 

8 

17 

25 

33 

42 

50 

59 

67 

75 

8 

92 

2 

3 

5 

7 

9 

17 

26 

34 

43 

51 

60 

69 

77 

7 

93 

2 

4 

5 

7 

9 

18 

26 

35 

44 

53 

61 

70 

79 

6 

94 

2 

4 

5 

7 

9 

18 

27 

36. 

45 

54 

63 

72 

81 

5 

95 

2 

4 

6 

7 

9 

18 

28 

37 

46 

55 

64 

73 

83 

4 

96 

2 

4 

6 

8 

9 

19 

28 

38 

47 

56 

66 

75 

84 

3 

97 

2 

4 

6 

8 

10 

19 

29 

38 

48 

57 

67 

77 

86 

2 

98 

2 

4 

6 

8 

10 

20 

29 

39 

49 

59 

69 

78 

88 

1 

99 

2 

4 

6 

8 

10 

20 

30 

40 

50 

60 

70 

80 

90 

0 

100 

2 

4 

6 

8 

10 

20 

31 

41 

51 

61 

71 

82 

92 


83 








































TABLE XI. POINT SWITCH TURNOUTS FROM STRAIGHT 

TRACK. 

Gage 4 feet 84 inches. Spread at heel 54 inches. 





<D 




1 l_ 


to 

M 



s 

6 

CD 

Tc 

c 

<4-4 _ 

6- 

’m 

A 

- 3 ^ 

° O © 

2 a > 

+3 
<4-4 3 

° o ci 
« a > 

03 

h-3 

e Ordi 

e, Out 

i Rail. 

r U 

Sh 

C 

Os-' 

£«* 

O 

T5.A 
o3 O 

«-*e 

5 

6 

O 

tH 

M 

O 

Sh 

c * 

o 

-4-3 . 

3 D 

p 5 3 

oint 

BF 

rrt 

.A A cc 

SO 

" a 

5 6 

4^0 O 

.a o£ 

o 4 ->mh 

M 

O 

i~ 



h-] 

32 

pH 

Q 

Ph 

3 

a 

o 

o 

pp 

Pp 


o / 


O f 

Feet. 

o / 

Feet. 

Ins. 

Ins. 

o / 




6 

9 314 

15 

1 45 

316.56 

18 11 

58.04 

m 

61 

12 59 

4.39 

41.31 

6 

64 

8 48 

15 


373.81 

15.22 

61.04 

84 

6# 

12 02 

4.75 

42.93 


7 

8 10 

15 


436.47 

13 09 

63.96 

8i 

6i 3 s 

11 13 

5.09 

44.48 

7 

7* 

7 374 

15 

(^ 

504.67 

11 22 

66.81 

8 

6 

10 304 

5.44 

45.98 

74 

8 

7 09 

15 

i i 

578.60 

9 55 

69.58 

71 

K1 3 

°ye 

9 544 

5.77 

47.37 

8 

84 

6 44 

15 

l i 

658.60 

8 42 

72.30 

74 

51 

9 22 

6.11 

48.74 

84 

9 

6 214 

15 


744.97 

7 42 

74.94 

7i 

5i 7 e - 

8 534 

6.43 

50.01 

9 

94 

6 014 

15 


837.94 

6 51 

77.53 

7 

51 

8 274 

6.76 

51.30 

94 

10 

5 434 

15 


937.82 

6 07 

80.05 

6f 

51 

8 04 

7.09 

52.52 

10 

8 

7 09 

18 

1 274 

567.50 

10 07 

74.44 

8 IS 

6» 

9 48 

5.83 

52.12 

8 

84 

6 44 

18 

(i 

644.27 

8 54 

77.35 

8i 

64 

9 154 

6.18 

53.66 

84 

9 

6 214 

18 


726.70 

7 53 

80.19 

8 

6 

8 464 

6.52 

55.12 

9 

94 

6 014 

18 

i i 

814.91 

7 02 

82.98 

71 

51 

8 204 

6.86 

56.56 

94 

10 

5 434 

18 


909.09 

6 18 

85.71 

7S 

514 

7 56 

7.21 

58.00 

10 

104 

5 27 

18 


1009.50 

5 41 

88.38 

71 

54 

7 36 

7.53 

59.29 

104 

11 

5 12 

18 


1116.33 

5 08 

90.99 

7i 

51 

7 17 

7.85 

60.55 

11 

114 

4 59 

18 


1229.72 

4 40 

93.56 

7 

51 

6 59 

8.19 

61.84 

114 

12 

4 464 

18 


1350.35 

4 15 

96.08 


54 

6 434 

8.51 

63.01 

12 

10 

5 434 

20 

1 19 

897.12 

6 23 

89.11 

8 

6 

7 53 

7.26 

61.33 

10 

104 

5 27 

20 


994.78 

5 46 

91.90 

7\% 

51 

7 324 

7.59 

62.71 

104 

11 

5 12 

20 

i i 

1098.38 

5 13 

94.63 

7f 

51 

7 134 

7.92 

64.06 

11 

114 

4 59 

20 


1207.98 

4 45 

97.31 

7 is 

5i 9 s 

6 56 

8.26 

65.44 

114 

12 

4 464 

20 


1324.18 

4 20 

99.96 

7i 

5tb 

6 40 

8.58 

66.71 

12 





Spread at Heel 5 inches. 






4 

14 15 

12 

1 594 

139.92 

41 53 

42.08 

9f 

7 is 

19 21 

2.93 

30.97 

4 

44 

12 41 

12 


178.07 

32 37 

45.34 

9, 7 B 

7 iTS 

17 164 

3.29 

32.83 

44 

5 

11 25 

15 

1 354 

218.69 

26 26 

52.63 

9f 

7i% 

15 31 

3.67 

38.69 

5 

54 

10 234 

15 

i t 

265.75 

21 41 

55.90 

94 

74 

14 094 

4.03 

40.56 

54 

0 

9 314 

15 


317.72 

18 07 

59.09 

94 

61 

13 004 

4.38 

42.37 

6 

64 

8 48 

15 

( i 

374.80 

15 20 

62.21 

$n 

614 

12 024 

4.74 

44.12 

64 

7 

8 10 

15 


437.11 

13 08 

65.25 

814 

64 

11 134 

5.09 

45.78 

7 

74 

7 374 

15 


504.75 

11 22 

68.23 

8/g 

6 is 

10 304 

5.44 

47.39 

74 

8 

7 09 

15 


577.88 

9 56 

71.13 

8 is 

64 

9 54 

5.77 

48.90 

8 

84 

6 44 

15 


656.80 

8 44 

73.97 

8 

6 

9 21 

6.11 

50.39 

84 

9 

6 214 

15 


741.72 

7 44 

76.75 

7| 

51 

8 53 

6.44 

51.77 

9 

94 

6 014 

15 


832.82 

6 53 

79.47 

74 

5| 

8 26 

6.78 

53.19 

94 

10 

5 434 

15 


930.35 

6 10 

82.13 

71 

54 

8 02 

7.12 

54.56 

10 


From Camp’s Notes on Track, by permission. 









































TABLEjXII. MIDDLE ORDINATES FOR CURVING RAILS 
IN INCHES. 


D 



Length of Rail Chord in 

Feet 



D 

10 

15 

20 

24 

26 

27 

28 

29 

30 

31 

32 

33 

O 

2 


£ 

£ 

1 

4 

3 

8 

3 

8 

3 

8 

1 

2 

£ 

1 

2 

£ 

5 

8 

O 

2 

3 

£ 

£ 

3. 

8 

£ 

£ 

S 

I 

5 

8 

3 

4 

A 

4 

S 

7 

8 

3 

4 

£ 

£ 

i 

5 

8 

3 

4 

s 

7 

8 

7 

8 

1 

1 

is 

1£ 

4 

5 

£ 

£ 

£ 

3 

4 

7 

8 

1 

1 

l£ 

U 

1£ 

is 

if 

5 

6 

S 

A 

8 

5 

7 

8" 

1 

l£ 

1£ 

IS 

if 

1£ 

is 

1 _3 

1 4 

6 

7 

£ 

S 

3. 

1 

1£ 

IS 

IS 

1£ 

1 5 
*8 

IS 

is 

2 

7 

8 

£ 

£ 

7_ 

~8 

It 

11 

l£ 

15 

1 8 

13. 

1 4 

IS 

2 

2£ 

2£ 

8 

9 


£ 

1 

IS 

1 5 
A 8 

IS 


2 

2£ 

2£ 

2S 

2| 

9 

10 


£ 

1 

1£ 

IS 

l£ 

2 

2£ 

2S 

2£ 

2S 

2S 

10 

12 

£ 

S 

1£ 

IS 

2£ 

2£ 

2£ 

2S 

2S 

3 

3£ 

3S 

12 

14 

S 

7 

1£ 

2 

2£ 

2S 

2£ 

3£ 

3£ 

3£ 

3S 

4 

14 

16 

£ 

1 

IS 

21 

2+ 

3 

3£ 

3£ 

3S 

4 

4£ 

4£ 

16 

18 

£ 

1 

1£ 

2S 

3£ 

3S 

3S 

4 

4£ 

4£ 

4S 

6£ 

18 

20 

£ 

i£ 

2 

3 

3£ 

3£ 

4£ 

4S 

4S 

5 

5S 

5S 

20 

25 

S 

1£ 

2S 

3S 

4S 

4S 

5£ 

5£ 

5S 

6£ 

6S 

7£ 

25 

30 

A 

is 

3£ 

44 

5£ 

5S 

6£ 

6S 

7£ 

7£ 

8 

8£ 

30 

35 

7 

2 

3S 

5£ 

6£ 

61 

7£ 

7S 

8£ 

8S 

9S 

10 

35 

40 

1 

2f 

4£ 

6 

7£ 

7S 

8£ 

8S 

9S 

10 

ios 

ns 

40 

45 

1£ 

2| 

4f 

6S 

8 

8£ 

9£ 

9£ 

10S 

ll£ 

12£ 

12S 

45 

50 

1£ 

3 

5£ 

7£ 

8£ 

9£ 

10£ 

11 

US 

12| 

13S 

14£ 

50 


TABLE XIII. ELEVATION OF OUTER RAIL IN INCHES. 


Velocity in Miles per Hour. 


















10 

15 

20 

25 

30 

35 

40 

45 

50 

55 

60 

65 

70 


O 

1 


£ 

£ 

3 

8 

S 

3 

4 

i£ 

If 

is 

2 

2S 

2S 

3£ 

O 

1 

2 

£ 

3 

8 

£ 

7 

■8 

1* 

IS 

2£ 

2-S 

3£ 

4 

4S 

54 

64 

2 

3 

£ 

£ 

3. 

1£ 

is 

2S 

3£ 

4 

4£ 

6 

7£ 

8S 

9S 

3 

4 

£ 

S 

1 

IS 

2S 

3£ 

4£ 

5S 

61 

8 

9£ 



4 

5 

S 

3 

4 

1£ 

2 

3 

4 

5£ 

61 

8£ 





5 

6 

3 

¥ 

1 

IS 

2£ 

3£ 

4S 

6£ 

8 






6 

7 

£ 

1£ 

li 

2S 

4£ 

5S 

7S 







7 

8 

£ 

1£ 

2£ 

3£ 

4S 

6£ 

8S 







8 

9 

S 

IS 

2S 

3S 

5S 

7£ 








9 

10 

A 

4 

i£ 

21 

4£ 

5S 

8£ 








10 

11 

S 

IS 

2S 

44 

64 

8| 








11 

12 

s 

IS 

3£ 

4£ 

7£ 









12 

13 

s 

2 

3S 

5S 

7S 









13 

14 

1 

2£ 

3S 

5S 

8S 









14 

15 

1 

2£ 

3£ 

6£ 

8S 









15 

16 

is 

2£ 

4£ 

6S 










16 

17 

1£ 

2f 

4£ 

7 










17 

18 

is 

2S 

4S 

7£ 










18 

19 

IS 

2£ 

5 

7S 










19 

20 

is 

3 

5£ 

8£ 










20 


85 































































TABLE XIV. CORRECTIONS TO HORIZONTAL DISTANCES FOR 

STADIA WORK. 

In feet per 100 ft. 


Vertical 

Angle. 

O' 

10' 

20' 

30' 

40' 

50' 

V ertical 
Angle. 

0° 




.01 

.01 

.02 

0° 

1 

.03 

.04 

.05 

.07 

.08 

.10 

1 

2 

.12 

.14 

.17 

.19 

.22 

.24 

2 

3 

.27 

.31 

.34 

.37 

.41 

.45 

3 

4 

.49 

.53 

.57 

.62 

.66 

.71 

4 

5 

.76 

.81 

.86 

.92 

.98 

1.03 

5 

6 

1.09 

1.15 

1.22 

1.28 

1.35 

1.42 

6 

7 

1.49 

1.56 

1.63 

1.70 

1.78 

1.86 

7 

8 

1.94 

2.02 

2.10 

2.18 

2.27 

2.36 

8 

9 

2.45 

2.54 

2.63 

2.72 

2.82 

2.92 

9 

10 

3.02 

3.12 

3.22 

3.32 

3.43 

3.53 

10 

11 

3.64 

3.75 

3.86 

3.97 

4.09 

4.21 

11 

12 

4.32 

4.44 

4.56 

4.68 

4.81 

4.93 

12 

13 

5.06 

5.19 

5.32 

5.45 

5.58 

5.72 

13 

14 

5.85 

5.99 

6.13 

6.27 

6.41 

6.55 

14 

15 

6.70 

6.84 

6.99 

7.14 

7.29 

7.44 

15 

16 

7.60 

7.75 

7.91 

8.07 

8.23 

8.39 

16 

17 

8.55 

8.71 

8.88 

9.04 

9.21 

9.38 

17 

18 

9.55 

9.72 

9.89 

10.07 

10.24 

10.42 

18 

19 

10.60 

10.78 

10.96 

11.14 

11.33 

11.51 

19 

20 

11.70 

11.89 

12.07 

12.26 

12.46 

12.65 

20 

21 

12.84 

13.04 

13.23 

13.43 

13.63 

13.83 

21 

22 

14.03 

14.24 

14.44 

14.64 

14.85 

15.06 

22 

23 

15.27 

15.48 

15.69 

15.90 

16.11 

16.33 

23 

24 

16.54 

16.76 

16.98 

17.20 

17.42 

17.64 

24 

25 

17.86 

18.08 

18.31 

18.53 

18.76 

18.99 

25 

26 

19.22 

19.45 

19.68 

19.91 

20.14 

20.38 

26 

27 

20.61 

20.85 

21.08 

21.32 

21.56 

21.80 

27 

28 

22.04 

22.28 

22.52 

22.77 

23.01 

23.26 

28 

29 

23.50 

23.75 

24.00 

24.25 

24.50 

24.75 

29 

30 

25.00 

25.25 

25.51 

25.76 

26.01 

26.27 

30 


86 





























TABLE XV. CUBIC YARDS PER 100 FEET. SLOPES i : 1. 


rd 

-u 

Q 

Base 

14 

Base 

16 

Base 

18 

Base 

20 

Base 

22 

Base 

26 

Base 

28 

Base 

30 

Base 

32 

1 

53 

60 

68 

75 

82 

97 

105 

112 

119 

2 

107 

122 

137 

152 

167 

196 

211 

226 

241 

3 

163 

186 

208 

230 

253 

297 

319 

342 

364 

4 

222 

252 

281 

311 

341 

400 

430 

459 

489 

5 

282 

319 

356 

394 

431 

505 

542 

579 

616 

6 

344 

389 

433 

478 

522 

611 

656 

700 

745 

7 

408 

460 

512 

564 

616 

719 

771 

823 

875 

8 

474 

533 

593 

652 

711 

830 

889 

948 

1007 

9 

542 

608 

675 

742 

808 

942 

1008 

1075 

1142 

10 

611 

685 

759 

833 

907 

1056 

1130 

1204 

1278 

11 

682 

764 

845 

927 

1008 

1171 

1253 

1334 

1416 

12 

756 

844 

933 

1022 

mi 

1289 

1378 

1467 

1556 

13 

831 

926 

1023 

1119 

1216 

1408 

1505 

1601 

1697 

14 

907 

1010 

1115 

1219 

1322 

1530 

1633 

1737 

1841 

15 

986 

1096 

1208 

1320 

1431 

1653 

1764 

1875 

1986 

16 

1067 

1184 

1304 

1422 

1541 

1778 

1896 

2015 

2133 

17 

1149 

1274 

1401 

1527 

1653 

1905 

2031 

2156 

2282 

18 

1233 

1366 

1500 

1633 

1767 

2033 

2167 

2300 

2433 

19 

1319 

1460 

1601 

1742 

1882 

2164 

2305 

2445 

2586 

20 

1407 

1555 

1704 

1852 

2000 

2296 

2444 

2593 

2741 

21 

1497 

1653 

1808 

1964 

2119 

2431 

2586 

2742 

2897 

22 

1589 

1752 

1915 

2078 

2241 

2567 

2730 

2893 

3056 

23 

1682 

1853 

2023 

2194 

2364 

2705 

2875 

3045 

3216 

24 

1778 

1955 

2133 

2311 

2489 

2844 

3022 

3200 

3378 

25 

1875 

2060 

2245 

2431 

2616 

2986 

3171 

3356 

3542 

26 

1974 

2166 

2359 

2552 

2744 

3130 

3322 

3515 

3707 

27 

2075 

2274 

2475 

2675 

2875 

3275 

3475 

3675 

3875 

28 

2178 

2384 

2593 

2800 

3007 

3422 

3630 

3837 

4044 

29 

2282 

2496 

2712 

2927 

3142 

3571 

3786 

4001 

4215 

30 

2389 

2610 

2833 

3056 

3278 

3722 

3944 

4167 

4389 

31 

2497 

2726 

2956 

3186 

3416 

3875 

4105 

4334 

4564 

32 

2607 

2844 

3081 

3319 

3556 

4030 

4267 

4504 

4741 

33 

2719 

2964 

3208 

3453 

3697 

4186 

4431 

4675 

4919 

34 

2833 

3085 

3337 

3589 

3841 

4344 

4596 

4848 

5100 

35 

2949 

3208 

3468 

3727 

3986 

4505 

4764 

5023 

5282 

36 

3067 

3333 

3600 

3867 

4133 

4667 

4933 

5200 

5467 

37 

3186 

3460 

3734 

4008 

4282 

4831 

5105 

5378 

5653 

38 

3307 

3589 

3870 

4152 

4433 

4996 

5278 

5559 

5841 

39 

3431 

3719 

4008 

4297 

4586 

5164 

5453 

5742 

6031 

40 

3556 

3852 

4148 

4445 

4741 

5333 

5630 

5926 

6222 

41 

3682 

3986 

4290 

4594 

4897 

5505 

5808 

6112 

6416 

42 

3811 

4122 

4433 

4744 

5056 

5678 

5989 

6300 

6611 

43 

3942 

4260 

4579 

4897 

5216 

5853 

6171 

6490 

6808 

44 

4074 

4400 

4726 

5052 

5378 

6030 

6356 

6682 

7007 

45 

4208 

4541 

4875 

5208 

5542 

6208 

6542 

6875 

7208 

46 

4344 

4684 

5026 

5367 

5707 

6389 

6730 

7070 

7411 

47 

4482 

4830 

5179 

5527 

5875 

6571 

6919 

7267 

7616 

48 

4622 

4978 

5333 

5689 

6044 

6756 

7111 

7467 

7822 

49 

4764 

5127 

5490 

5853 

6216 

6942 

7305 

7668 

8031 

50 

4907 

5278 

5648 

6019 

6389 

7130 

7500 

7870 

8241 

51 

5053 

5430 

5808 

6186 

6564 

7319 

7697 

8075 

8453 

52 

5200 

5584 

5970 

6356 

6741 

7511 

7896 

8281 

8667 

53 

5349 

5741 

6134 

6527 

6919 

7705 

8097 

8490 

8883 

54 

5500 

5900 

6300 

6700 

7100 

7900 

8300 

8700 

9100 

55 

5653 

6060 

6468 

6875 

7282 

8097 

8505 

8912 

9319 

56 

5807 

6222 

6637 

7052 

7467 

8296 

8711 

9126 

9541 

57 

5964 

6386 

6808 

7231 

7653 

8497 

8919 

9342 

9764 

58 

6122 

6552 

6981 

7411 

7841 

8700 

9130 

9559 

9989 

59 

6282 

6719 

7156 

7593 

8031 

8905 

9342 

9779 

10216 

60 

6444 

6889 

7333 

7778 

8222 

9111 

9556 

10000 

19445 


87 
























TABLE XV. CUBIC YARDS PER 100 FEET. SLOPES 


Depth, 

d. 

Base. 

14 

Base 

16 

Base 

18 

1 

54 

61 

69 

2 

111 

126 

141 

3 

172 

194 

217 

4 

237 

267 

296 

5 

306 

343 

380 

6 

378 

422 

467 

7 

454 

506 

557 

8 

533 

593 

652 

9 

617 

683 

750 

10 

704 

778 

852 

11 

794 

876 

957 

12 

889 

978 

1067 

13 

987 

1083 

1180 

14 

1089 

1193 

1296 

15 

1194 

1306 

1417 

16 

1304 

1422 

1541 

17 

1417 

1543 

1669 

18 

1533 

1667 

1800 

19 

1654 

1794 

1935 

20 

1778 

1926 

2074 

21 

1906 

2061 

2217 

22 

2037 

2200 

2363 

23 

2172 

2343 

2513 

24 

2311 

2489 

2667 

25 

2454 

2639 

2824 

26 

2600 

2793 

2985 

27 

2750 

2950 

3150 

28 

2904 

3111 

3319 

29 

3061 

3276 

3491 

30 

3222 

3444 

3667 

31 

3387 

3617 

3846 

32 

3556 

3793 

4030 

33 

3728 

3972 

4217 

34 

3904 

4156 

4407 

35 

4083 

4343 

4602 

36 

4267 

4533 

4800 

37 

4454 

4728 

5002 

38 

4644 

4926 

5207 

39 

4839 

5128 

5417 

40 

5037 

5333 

5630 

41 

5239 

5543 

5846 

42 

5444 

5756 

6067 

43 

5654 

5972 

6291 

44 

5867 

6193 

6519 

45 

6083 

6417 

6750 

46 

6304 

6644 

6985 

47 

6528 

' 6876 

7224 

48 

6756 

7111 

7467 

49 

6987 

7350 

7713 

50 

7222 

7593 

7963 

51 

7461 

7839 

8217 

52 

7704 

8089 

8474 

53 

7950 

8343 

8735 

54 

8200 

8600 

9000 

55 

8454 

8861 

9269 

56 

8711 

9126 

9541 

57 

8972 

9394 

9817 

58 

9237 

9667 

10096 

59 

9506 

9943 

10380 

60 

9778 

10222 

10667 


Base 

20 

Base 

22 

Base 

28 

Base 

30 

76 

83 

106 

113 

156 

170 

215 

230 

239 

261 

328 

350 

326 

356 

444 

474 

417 

454 

565 

602 

511 

556 

689 

733 

609 

661 

817 

869 

711 

770 

948 

1007 

817 

883 

1083 

1150 

926 

1000 

1222 

1297 

1039 

1120 

1365 

1447 

1156 

1244 

1511 

1600 

1276 

1372 

1661 

1757 

1400 

1504 

1815 

1919 

1528 

1639 

1972 

2083 

1659 

1779 

2133 

2252 

1794 

1921 

2298 

2424 

1933 

2067 

2467 

2600 

2076 

2217 

2639 

2780 

2222 

2370 

2815 

2963 

2372 

2528 

2994 

3150 

2526 

2689 

3178 

3341 

2683 

2854 

3365 

3535 

2844 

3022 

3556 

3733 

3009 

3194 

3750 

3935 

3178 

3370 

3948 

4141 

3350 

3550 

4151 

4350 

3526 

3733 

4356 

4563 

3706 

3920 

4565 

4780 

3889 

4111 

4778 

5000 

4076 

4306 

4994 

5224 

4267 

4504 

5215 

5452 

4461 

4706 

5439 

5683 

4659 

4911 

5667 

5919 

4861 

5120 

5898 

6157 

5067 

5333 

6133 

6400 

5276 

5550 

6372 

6646 

5489 

5770 

6615 

6896 

5706 

5994 

6861 

7150 

5926 

6222 

7111 

7408 

6150 

6454 

7365 

7669 

6378 

6689 

7622 

7933 

6609 

6928 

7883 

8202 

6844 

7170 

8148 

8474 

7083 

7417 

8417 

8750 

7326 

7667 

8689 

9030 

7572 

7920 

8965 

9313 

7822 

8178 

9244 

9600 

8076 

8439 

9528 

9891 

8333 

8704 

9815 

10185 

8594 

8972 

10106 

10483 

8859 

9244 

10400 

10785 

9128 

9520 

10698 

11091 

9400 

9800 

11000 

11400 

9676 

10083 

11306 

11713 

9956 

10370 

11615 

12030 

10239 

10661 

11928 

12350 

10526 

10956 

12244 

12674 

10817 

11254 

12565 

13002 

mil 

11556 

12889 

13333 


4:1. 


Base 

32 


120 

244 

372 

504 

639 

778 

920 

1067 

1217 

1370 

1528 

1689 

1854 

2022 

2194 

2370 

2550 

2733 

2920 

3111 

3305 

3504 

3706 

3911 

4120 

4332 

4550 

4771 

4994 

5222 

5454 

5689 

5928 

6170 

6417 

6667 

6920 

7178 

7439 

7704 

7972 

8244 

8520 

8800 

9083 

9370 

9661 

9956 

10254 

10556 

10861 

11170 

11483 

11800 

12121 

12445 

12772 

13104 

13439 

13778 


88 






















TABLE XV. CUBIC YARDS PER 100 FEET. SLOPE 1:1. 


. 

Q 

Base 

14 

Base 

16 

Base 

18 

Base 

20 

Base 

22 

Base 

28 

Base 

30 

Base 

32 

1 

56 

63 

70 

78 

85 

107 

115 

122 

2 

119 

133 

148 

163 

178 

222 

237 

252 

3 

189 

211 

233 

256 

278 

344 

367 

389 

4 

267 

296 

326 

356 

385 

474 

504 

533 

5 

352 

389 

426 

463 

500 

611 

648 

685 

6 

444 

489 

533 

578 

622 

756 

800 

844 

7 

544 

596 

648 

700 

752 

907 

959 

1011 

8 

652 

711 

770 

830 

889 

1067 

1126 

1185 

9 

767 

833 

900 

967 

1033 

1233 

1300 

1367 

10 

889 

963 

1037 

1111 

1185 

1407 

1481 

1556 

11 

1019 

1100 

1181 

1263 

1344 

1589 

1670 

1752 

12 

1156 

1244 

1333 

1422 

1511 

1778 

1867 

1956 

13 

1300 

1396 

1493 

1589 

1685 

1974 

2070 

2167 

14 

1452 

1556 

1659 

1763 

1867 

2178 

2281 

2385 

15 

1611 

1722 

1833 

1944 

2056 

2389 

2500 

2611 

16 

1778 

1896 

2015 

2133 

2252 

2607 

2726 

2844 

17 

1952 

2078 

2204 

2330 

2456 

2833 

2959 

3085 

18 

2133 

2267 

2400 

2533 

2667 

3067 

3200 

3333 

19 

2322 

2463 

2604 

2744 

2885 

3307 

3448 

3589 

20 

2519 

2667 

2815 

2963 

3111 

3556 

3704 

3852 

21 

2722 

2878 

3033 

3189 

3344 

3811 

3967 

4122 

22 

2933 

3096 

3259 

3422 

3585 

4074 

4237 

4400 

23 

3152 

3322 

3493 

3663 

3833 

4344 

4515 

4685 

24 

3378 

3556 

3733 

3911 

4089 

4622 

4800 

4978 

25 

3611 

3796 

3981 

4167 

4352 

4907 

5093 

5278 

26 

3852 

4044 

4237 

4430 

4622 

5200 

5393 

5585 

27 

4100 

4300 

4500 

4700 

4900 

5500 

5700 

5900 

£8 

4356 

4563 

4770 

4978 

5185 

5807 

6015 

6222 

29 

4619 

4833 

5048 

5263 

5478 

6122 

6337 

6552 

30 

4889 

5111 

5333 

5556 

5778 

6444 

6667 

6889 

31 

5167 

5396 

5626 

5856 

6085 

6774 

7004 

7233 

32 

5452 

5689 

5926 

6163 

6399 

7111 

7348 

7585 

33 

5744 

5989 

6233 

6478 

6722 

7456 

7700 

7944 

34 

6044 

6296 

6548 

6800 

7052 

7807 

8059 

8311 

35 

6352 

6611 

6870 

7130 

7389 

8167 

8426 

8685 

36 

6667 

6933 

7200 

7467 

7733 

8533 

8800 

9067 

37 

6989 

7263 

7537 

7811 

8085 

8907 

9181 

9456 

38 

7319 

7600 

7881 

8163 

8444 

9289 

9570 

9852 

39 

7656 

7944 

8233 

8522 

8811 

9678 

9967 

10256 

40 

8000 

8296 

8593 

8889 

9185 

10074 

10370 

10667 

41 

8352 

8656 

8959 

9263 

9567 

10478 

10781 

11085 

42 

8711 

9022 

9333 

9644 

9956 

10889 

11200 

11511 

43 

9078 

9396 

9715 

10033 

10352 

11307 

11626 

11944 

44 

9452 

9778 

10104 

10430 

10756 

11733 

12059 

12385 

45 

9833 

10167 

10500 

10833 

11167 

12167 

12500 

12833 

46 

10222 

10563 

10904 

11244 

11585 

12607 

12948 

13289 

47 

10619 

10967 

11315 

11663 

12011 

13056 

13404 

13752 

48 

11022 

11378 

11733 

12089 

12444 

13511 

13867 

14222 

49 

11433 

11796 

12159 

12522 

12885 

13974 

14337 

14700 

50 

11852 

12222 

12593 

12963 

13333 

14444 

14815 

15185 

51 

12278 

12656 

13033 

13411 

13790 

14922 

15300 

15678 

52 

12711 

13096 

13481 

13867 

14252 

15407 

15793 

16178 

53 

13152 

13544 

13937 

14330 

14722 

15900 

16293 

16685 

54 

13600 

14000 

14400 

14800 

15200 

16400 

16800 

17200 

55 

14056 

14463 

14870 

15278 

15685 

16907 

17315 

17722 

56 

14519 

14933 

15348 

15763 

16178 

17422 

17837 

18252 

57 

14989 

15411 

15833 

16256 

16678 

17944 

18367 

18789 

58 

15467 

15896 

16326 

16756 

17185 

18474 

18904 

19333 

59 

15952 

16389 

16826 

17263 

17700 

19011 

19448 

19885 

60 

16444 

16889 

17333 

17778 

18222 

19556 

20000 

20244 


% 


89 




























































TABLE XV. CUBIC YARDS PER 100 FEET. SLOPES 1£ : 1. 


Depth, 

d. 

Base 

14 

Base 

16 

Base 

18 

Base 

20 

Base 

22 

Base 

28 

Base 

30 

Base 

32 


1 

57 

65 

72 

80 

87 

109 

117 

124 


2 

126 

141 

156 

170 

185 

230 

244 

259 


3 

206 

228 

250 

272 

294 

361 

383 

406 


4 

296 

326 

356 

385 

415 

504 

533 

563 


5 

398 

435 

472 

509 

546 

657 

694 

731 


6 

511 

556 

600 

644 

689 

822 

867 

911 


7 

635 

687 

739 

791 

843 

998 

1050 

1102 


8 

770 

830 

889 

948 

1007 

1185 

1244 

1304 


9 

917 

983 

1050 

1116 

1183 

1383 

1450 

1517 


10 

1074 

1148 

1222 

1296 

1370 

1593 

1667 

1741 


11 

1243 

1324 

1406 

1487 

1569 

1813 

1894 

1976 


12 

1422 

1511 

1600 

1689 

1778 

2044 

2133 

2222 


13 

1613 

1709 

1806 

1902 

1998 

2287 

2383 

2480 


14 

1815 

1919 

2022 

2126 

2230 

2541 

2644 

2748 


15 

2028 

2139 

2250 

2361 

2472 

2806 

2917 

3028 


16 

2252 

2370 

2489 

2607 

2726 

3081 

3200 

3319 


17 

2487 

2613 

2739 

2865 

2991 

3369 

3494 

3620 


18 

2733 

2867 

3000 

3133 

3267 

3667 

3800 

3933 


19 

2991 

3131 

3272 

3413 

3554 

3976 

4117 

4257 


20 

3259 

3407 

3556 

3704 

3852 

4296 

4444 

4592 


21 

3539 

3694 

3850 

4005 

4161 

4628 

4783 

4939 


22 

3830 

3993 

4156 

4318 

4482 

4970 

5133 

5296 


23 

4131 

4302 

4472 

4642 

4813 

5324 

5494 

5665 


24 

4444 

4622 

4800 

4978 

5156 

5689 

5867 

6044 


25 

4769 

4954 

5139 

5324 

5509 

6065 

6250 

6435 


26 

5104 

5296 

5489 

5681 

5874 

6452 

6644 

6837 


27 

5450 

5650 

5850 

6050 

6250 

6850 

7050 

7250 


28 

5807 

6015 

6222 

6430 

6637 

7259 

7467 

7674 


29 

6176 

6391 

6606 

6820 

7035 

7680 

7894 

8109 * 


30 

6556 

6778 

7000 

7222 

7445 

8111 

8333 

8555 


31 

6946 

7176 

7406 

7635 

7865 

8554 

8783 

9013 


32 

7348 

7585 

7822 

8059 

8296 

9007 

9244 

9'482 


33 

7761 

8006 

8250 

8494 

8739 

9472 

9717 

9962 


34 

8135 

8437 

8689 

8941 

9193 

9948 

10200 

10452 


35 

8620 

8880 

9139 

9398 

9657 

10435 

10694 

10954 


36 

9067 

9333 

9600 

9867 

10134 

10933 

11200 

11467 


37 

9524 

9798 

10072 

10346 

10621 

11443 

11717 

11991 


38 

9993 

10274 

10556 

10837 

11119 

11963 

12244 

12526 


39 

10472 

10761 

11050 

11339 

11628 

12494 

12783 

13072 


40 

10963 

11259 

11556 

11852 

12148 

13037 

13333 

13630 


41 

11465 

11769 

12072 

12376 

12680 

13591 

13894 

14198 


42 

11978 

12289 

12600 

12911 

13223 

14156 

14467 

14778 


43 

12502 

12820 

13139 

13457 

13776 

14731 

15050 

15369 


44 

13037 

13363 

13689 

14015 

14341 

15319 

15644 

15970 


45 

13583 

13917 

14250 

14583 

14917 

15917 

16250 

16583 


46 

14141 

14481 

14822 

15163 

15504 

16526 

16867 

17207 


47 

14709 

15057 

15406 

15754 

16102 

17146 

17494 

1 784? 


48 

15289 

15644 

16000 

16356 

16711 

17778 

18133 

18489 


49 

15880 

16243 

16606 

16968 

17382 

18420 

18783 

19146 


50 

16481 

16852 

17222 

17592 

17963 

19074 

19444 

19815 


51 

17094 

17472 

17850 

18228 

18605 

19739 

20117 

20494 


52 

17719 

18104 

18489 

18874 

19259 

20415 

20800 

21185 


53 

18354 

18746 

19139 

19531 

19924 

21102 

21494 

21887 


54 

19000 

19400 

19800 

20200 

20600 

21800 

22200 

22600 


55 

19657 

20065 

20472 

20880 

21287 

22509 

22917 

23324 


56 

20326 

20741 

21156 

21570 

21986 

23230 

23644 

24059 


57 

21006 

21428 

21850 

22272 

22695 

23961 

24383 

24805 


58 

21696 

22126 

22556 

22985 

23415 

24704 

25133 

25563 


59 

22398 

22835 

23272 

23709 

24147 

25457 

25894 

26332 


60 

23111 

23556 

24000 

24444 

24889 

26222 

26667 

27111 





90 




















































TABLE XV. CUBIC YARDS PER 100 FEET. SLOPES 2:1. 


Depth, 

d. 

Base 

14 

Base 

16 

Base 

18 

Base 

20 

Base 

22 

Base 

28 

Base 

30 

Base 

32 

1 

59 

67 

74 

81 

89 

111 

119 

126 

2 

133 

148 

163 

178 

193 

237 

252 

267 

3 

222 

244 

267 

289 

311 

378 

400 

422 

4 

326 

356 

385 

415 

445 

533 

563 

593 

5 

444 

481 

519 

556 

593 

704 

741 

778 

6 

578 

622 

667 

711 

756 

889 

933 

978 

7 

726 

778 

830 

881 

933 

1089 

1141 

1193 

8 

889 

948 

1007 

1067 

1126 

1304 

1363 

1422 

9 

1067 

1133 

1200 

1267 

1333 

1533 

1600 

1667 

10 

1259 

1333 

1407 

1481 

1556 

1778 

1852 

1926 

11 

1467 

1548 

1630 

1711 

1793 

2037 

2119 

2200 

12 

1689 

1778 

1867 

1956 

2044 

2311 

2400 

2489 

13 

1926 

2022 

2119 

2215 

2311 

2600 

2696 

2793 

14 

2178 

2281 

2385 

2489 

2593 

2904 

3007 

3111 

15 

2444 

2556 

2667 

2778 

2889 

3222 

3333 

3444 

16 

2726 

2844 

2963 

3081 

3200 

3556 

3674 

3793 

17 

3022 

3148 

3274 

3400 

3526 

3904 

4030 

4156 

18 

3333 

3467 

3600 

3733 

3867 

4267 

4400 

4533 

19 

3659 

3800 

3941 

4081 

4222 

4644 

4785 

4926 

20 

4000 

4148 

4296 

4444 

4593 

5037 

5185 

5333 

21 

4356 

4511 

4667 

4822 

4978 

5444 

5600 

5756 

22 

4726 

4889 

5052 

5215 

5378 

5867 

6030 

6193 

23 

5111 

5281 

5452 

5622 

5793 

6304 

6474 

6644 

24 

5511 

5689 

5867 

6044 

6222 

6756 

6933 

7111 

25 

5926 

6111 

6296 

6481 

6667 

7222 

7407 

7593 

26 

6356 

6548 

6741 

6933 

7126 

7704 

7896 

8089 

27 

6800 

7000 

7200 

7400 

7600 

8200 

8400 

8600 

28 

7259 

7467 

7674 

7881 

8089 

8711 

8919 

9126 

29 

7733 

7948 

8163 

8378 

8593 

9237 

9452 

9667 

30 

8222 

8444 

8667 

8889 

9111 

9778 

10000 

10222 

31 

8726 

8956 

9185 

9415 

9645 

10333 

10563 

10793 

32 

9244 

9482 

9719 

9956 

10193 

10904 

11141 

11378 

33 

9778 

10022 

10267 

10511 

10756 

11489 

11733 

11978 

34 

10326 

10578 

10830 

11081 

11333 

12089 

12341 

12593 

35 

10889 

11148 

11407 

11667 

11926 

12704 

12963 

13222 

36 

11467 

11733 

12000 

12267 

12533 

13333 

13600 

13867 

37 

12059 

12333 

12607 

12881 

13155 

13978 

14252 

14526 

38 

12667 

12948 

13230 

13511 

13792 

14637 

14919 

15200 

39 

13289 

13578 

13867 

14156 

14444 

15311 

15600 

15889 

40 

13926 

14222 

14519 

14815 

15111 

16000 

16296 

16593 

41 

14578 

14881 

15185 

15489 

15793 

16704 

17007 

17311 

42 

15244 

15556 

15867 

16178 

16489 

17422 

17733 

18044 

43 

15926 

16244 

16563 

16881 

17200 

18156 

18474 

18793 

44 

16622 

16948 

17274 

17600 

17926 

18904 

19230 

19556 

45 

17333 

17667 

18000 

18333 

18667 

19667 

20000 

20333 

46 

18059 

18400 

18741 

19081 

19422 

20444 

20785 

21126 

47 

18800 

19148 

19496 

19844 

20193 

21237 

21585 

21933 

48 

19556 

19911 

20267 

20622 

20978 

22044 

22400 

22756 

49 

20326 

20689 

21052 

21415 

21778 

22867 

23230 

23593 

50 

21111 

21481 

21852 

22222 

22593 

23704 

24074 

24444 

51 

21911 

22289 

22667 

23044 

23422 

24556 

24933 

25311 

52 

22726 

23111 

23496 

23881 

24267 

25422 

25807 

26193 

53 

23556 

23948 

24341 

24733 

25126 

26304 

26696 

27089 

54 

24400 

24800 

25200 

25600 

26000 

27200 

27600 

28000 

55 

25259 

25667 

26074 

26481 

26889 

28111 

28519 

28926 

56 

26133 

26548 

26963 

27378 

27793 

29037 

29452 

29867 

57 

27022 

27444 

27867 

28289 

28711 

29978 

30400 

30822 

58 

27926 

28356 

28785 

29215 

29645 

30933 

31363 

31793 

59 

28844 

29281 

29719 

30156 

30593 

31904 

32341 

32778 

60 

29778 

30222 

30667 

31111 

31556 

32889 

33333 

33778 





















TABLE XVI. FORMULAS. 


Circular Functions. 

Second quadrant: 

sin A = cos (A — 90) cos A = — sin (A — 90) 
tan A = — cot (A — 90) 

Third quadrant: 

sin A = — sin (A — 180) cos A = — cos (A — 180) 

tan A = tan (A — 180) 

Fourth quadrant: 

sin A = — cos (A — 270) cos A = sin (A — 270) 
tan A = — cot (A — 270) 


Right Triangle. 


sin A = - 
c 


cos A = - 
c 


tan A = - 
b 


c 2 = a 2 + 6 2 Area = ^ ab, where c is the hypotenuse. 
Oblique Triangle. 


sin A 
a 


sin B sin C 


b c 

a 2 = b 2 + c 2 — 2 be cos A 

tan \ (A - B) = tan^(A + 5) MA + 5) = 90-i C 

A - * (A + 5) + i (A - 5) 

B = * (A + B) - KA - 5) 

, . cos £ (A + B) , sin £ (A + £) 

c = (a +b) - = (a — 6) ——f-7-7- 

cos % (A — B) v sin ^ (A — B) 

tan £ A = i/ —— ^ ^ C \ where s = £ (a + b + c) 

S CL) 


sin A = 


_ 2Vs (s—a) (s —6) (s —c) 


6c 


, 2 (s—6) (s—c) 

be 


Area = % ab sin C = Vs (s — a) (s — 6) (s — c). 


General Formulas. 


sin A - -— = VT — cos 2 A = tan A cos A 

cosec A 

= 2 sin \ A cos h A = vers A cot ^ A 
= vers 2 A = V| (1 — cos 2 A). 


92 


















TABLE XVI. FORMULAS 


cos A — 


tan A 


vers A 


——- = \/1 — sin 2 A = cot A sin A 
sec A 

1 — vers A = 2 cos 2 jA — 1 = 1— 2 sin 
cos 2 % A — sin 2 %A = \/% — 

1 sin A. /——-- t / 1 


2 1 


A 


b cos 2 A 

-.' = \/sec 2 A — 1 = v 

cot A cos A “ ' 

sin 2 A 1 — cos 2 A 


cos 2 A 


- 1 


1 + cos 2 A 


exsec A 


sin 2 A 

1 — cos A = sin A tan \ A = 
= exsec A cos A 

sec A — 1 = tan A tan b A = 


= exsec A cot \ A. 
2 sin 2 i A 

vers A 


sin f 




cos A 


vers A 


2 


sin 2 A =2 sin A cos A 

V 


cos b A 


^1 + cos A 


T 

cos 2 A = 2 cos 2 A — 1 = cos 2 A — sin 2 A = 1 — 2 sin 2 A 

tan A . , . 1 — cos A 

= cosec A — cot A = -;—— 

sm A 


tan h A = -—■—-—^ 
2 1 + sec A 


-V 


1 — cos A 
1 + cos A 


tan 2 A = 
exsec \ A = 
exsec 2 A = 


2 tan A 
1 — tan 2 A 

1 — cos A 

1 + cos A + V2 (1 + cos A) 
2 tan 2 A 


1 - tan 2 A 

sin (A ± B) = sin A cos B ± sin B cos A 

cos {A + B) = cos A cos B =F sin A sin B 

sin A + sin B = 2 sin b (A A B) cos b (A — B) 

sin A — sin B = 2 cos ^ (A + S) sin ^ (A — B) 

cos A + cos B = 2 cos (A + £) cos ^ (A — B) 

cos A — cos B = — 2 sin ^ (A + B) sin b (A - B) 

sin 2 A — sin 2 B = cos 2 B — cos 2 A 

= sin (A + B) sin (A — B) 

cos 2 A — sin 2 B = cos (A + j3) cos (A — 5) 

sin (A + B) 

tan A + tan B = - — A -^ 

cos A cos B 

_ sin (A — 5) 

tan A — tan B = - s -n • 

cos A cos B 


93 































natural sines and cosines, 



0 

o 

1 

o 

2 

O 

3 

O 

4 ° 



Sine 

Cosin 

Sine 

Cosin 

Sin© 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

/ 

0 

.00000 

One. 

.01745 

.99985 

.03490 

.99939 

.05234 

799863 

.06976 

.99756 

60 

1 

.00029 

One. 

.01774 

.99984 

.03519 

.99938 

.05263 

.99861 

.07005 

.99754 

59 

2 

.00058 

One. 

.01803 

.99984 

.03548 

.99937 

.05292 

.99860 

.07034 

.99752 

58 

3 

.0ou8y 

One. 

.01832 

.99983 

.03577 

.99936 

.05321 

.99858 

.07063 

.99750 

57 

4 

.00116 

One. 

.01862 

.99983 

.03606 

.99935 

.05350 

.99857 

.07092 

.99748 

56 

5 

.00145 

[One. 

.01891 

.99982 

.03685 

.99934 

.05379 

.99855 

.07121 

.99746 

55 

6 

.00175 

One. 

.01920 

.99982 

.03664 

.99933 

.05408 

.99854 

.07150 

.99744 

54 

7 

.00204 

One. 

.01949 

.99981 

.03693 

.99932 

.05437 

.99852 

.07179 

.99742 

53 

8 

.00233 

One. 

.01978 

.99980 

.03723 

.99931 

.05466 

.99851 

.07208 

.99740 

52 

9 

.00262 

i_One. 

.02007 

.99980 

.03752 

.99930 

.05495 

.99849 

.07237 

.99738 

51 

10 

.00291 

One. 

.02036 

.99979 

.03781 

.99929 

.05524 

.99847 

.07266 

.99736 

50 

11 

.00320 

.99999 

.02065 

.99979 

.03810 

.99927 

.05553 

.99846 

.07295 

.99734 

49 

12 

.00349 

.99999 

.02094 

.99978 

.03839 

.99926 

.05582 

.99844 

.07324 

.99731 

48 

13 

.00378 

.99999 

.02123 

.99977 

.03868 

.99925 

.05611 

.99842 

.07353 

.99729 

47 

14 

.00407 

.99999 

.02152 

.99977 

.03897 

.99924 

.05640 

.99841 

.07382 

.99727 

46 

15 

.00436 

.99999 

.02181 

.99976 

.03926 

.99923 

.05669 

.99839 

.07411 

.99725 

45 

16 

.00465 

.99999 

.02211 

.99976 

.03955 

.99922 

.05698 

.99838 

.07440 

.99723 

44 

17 

.00495 

.99999 

.02240 

.99975 

.03984 

.99921 

.05727 

.99836 

.07469 

.99721 

43 

18 

.00524 

.99999 

.02269 

.99974 

.04013 

.99919 

.05756 

.99834 

.07498 

.99719 

42 

19 

.00553 

.99998 

.02298 

.99974 

.04042 

.99918 

.05785 

.99833 

.07527 

.99716 

41 

20 

.00582 

.99998 

.02327 

.99973 

.04071 

.99917 

.05814 

.99831 

.07556 

.99714 

40 

21 

.00611 

.99998 

.02356 

.99972 

.04100 

.99916 

.05844 

.99829 

.07585 

.99712 

39 

22 

.00640 

.99998 

.02385 

.99972 

.04129 

.99915 

.05073 

.99827 

.07614 

.99710 

38 

23 

.00669 

.99998 

.02414 

.99971 

.04159 

.99913 

.05902 

.99826 

.07643 

.99708 

37 

24 

.00698 

.99998 

.02443 

.99970 

.04188 

.99912 

.05931 

.99824 

.07672 

.99705 

36 

25 

.00727 

.99997 

.02472 

.99969 

04217 

.99911 

.05960 

.99822 

.07701 

.99703 

35 

26 

.00756 

.99997 

.02501 

.99969 

.04246 

.99910 

.05989 

.99821 

.07730 

.99701 

34 

27 

.00785 

.99997 

.02530 

.99968 

.04275 

.999C9 

.06018 

.99819 

.07759 

.99699 

33 

28 

.00814 

.99997 

.02560 

.99967 

.04304 

.99907 

.06047 

.99817 

.07788 

.99696 

32 

29 

.00844 

.99996 

.02589 

.99966 

.04333 

.99906 

.06076 

.99815 

.07817 

.99694 

31 

30 

.00873 

.99996 

.02618 

.99966 

.04362 

.99905 

.06105 

.99813 

.07846 

.99692 

30 

31 

.00902 

.99996 

.02647 

.99965 

.04391 

.99904 

.06134 

.99812 

.07875 

.99689 

29 

32 

.00931 

.99996 

.02676 

.93964 

.04420 

.99902 

.0G1C3 

.99810 

.07904 

.99687 

28 

33 

.00960 

.99995 

.02705 

.99963 

.04449 

.99901 

.06192 

.99808 

.07933 

.99685 

27 

34 

.00989 

.99995 

.02734 

.99963 

.04478 

.99900 

.06221 

.99806 

.07962 

.99683 

26 

35 

.01018 

.99995 

.02703 

.99962 

.04507 

.99898 

.06250 

.99804 

.07991 

.99680 

25 

36 

.01047 

.99995 

.02792 

.99901 

.04536 

.99897 

.06279 

.99803 

.08020 

.99678 

24 

37 

.01076 

.99994 

.02821 

.99960 

.04565 

.99896 

.06308 

.99801 

.08049 

.99676 

23 

38 

.01105 

.99994 

.02850 

.99959 

.04594 

.99894 

.06337 

.99799 

.08078 

.99673 

22 

39 

.01134 

.99994 

.02879 

.99959 

.04623 

.99893 

.06366 

.99797 

.08107 

.99671 

21 

40 

.01164 

.99993 

.02908 

.99958 

.04653 

.99892 

.06395 

.99795 

.08136 

.99668 

20 

41 

.01193 

.99993 

.02938 

.99957 

.04682 

.99890 

.06424 

.99793 

.08165 

.99666 

19 

42 

.01222 

.99993 

.02967 

.99956 

.04711 

.99889 

.06453 

.99792 

.08194 

.99664 

18 

43 

.01251 

.99992 

.02996 

.99955 

.04740 

.99888 

.06482 

.99790, 

.08223 

.99661 

17 

44 

.01280 

.99992 

.03025 

.99954 

.04769 

.99886 

.06511 

.99788 

.08252 

.99659 

16 

45 

.01309 

.99991 

.03054 

.99953 

.04798 

.99885 

.06540 

.99786 

.08281 

.99657 

15 

46 

.01338 

.99991 

.03083 

.99952 

.04827 

.99883 

.06569 

.99784 

.08310 

.99654 

14 

47 

.01367 

.99991 

.03112 

.99952 

.04856 

.99882 

.06598 

.99782 

.08339 

.99652 

13 

48 

.01396 

.99990 

.03141 

.99951 

.04885 

.99881 

.06627 

.99780 

.08368 

.99649 

12 

49 

.01425 

.99990 

.03170 

.99950 

.04914 

.99879 

.06656 

.99778 

.08397 

.99647 

11 

50 

.01454 

.99989 

.03199 

.99949 

.04943 

.99878 

.06685 

.99776 

.08426 

.99644 

10 

51 

.01483 

.99989 

.03228 

.99948 

.04972 

.99876 

.06714 

.99774 

.084551 

.99642 

9 

52 

.01513 

.99989 

.03257 

.99947 

.05001 

.99875 

.06743 

.99772 

.08484 

.99639 

8 

53 

.01542 

.99988 

.03286 

.99946 

.05030 

.99873 

.06773 

.99770 

.08513 

.99637 

7 

54 

.01571 

.99988 

.03316 

.99945 

.05059 

.99872 

.06802 

.99768 

.08542 

.99635 

6 

55 

.01600 

.99987 

.03345 

.99944 

.05088 

.99870 

.06831 

.99766 

.08571 

.99632 

5 

56 

.01629 

.99987 

.03374 

.99943 

.05117 

.99869 

.06860 

.99764 

.08600 

.99630 

4 

57 

.01658 

.99986 

.03403 

.99942 

.05146 

.99867 

.06889 

.99762 

.08629 

.99627 

3 

58 

.01687 

.99986 

.03432 

.99941 

.05175 

.99866 

.06918 

.99760 

.08058 

.99625 

2 

59 

.01716 

.99985 

.03461 

.99940 

.05205 

.99864 

.06947 

.99758 

.08687 

.99622 

1 

60 

.01745 

.99985 

.03490 

.99939 

.05234 

.99863 

.06976 

.99756 

.08716 

.99619 

0 

/ 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

$ 


89 ° 

O 

oo 

00 

o 

J> 

OO 

86 ° 

85 

* 



73 ^ 






















































NATURAL SINES AND COSINES. 



s 


e 

>° 

1 

r° 

8® 

9° 



Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

9 

0 

.08716 

.99619 

.10453 

.99452 

.12187 

.99255 

.13917 

.99027 

.15643 

.98769 

60 

1 

.08745 

.99617 

.10482 

.99449 

.12216 

.99251 

.13946 

.99023 

.15672 

.98764 

59 

2 

.08774 

.99614 

.10511 

.99446 

.12245 

.99248 

.13975 

.99019 

.15701 

.98^60 

58 

3 

.08803 

..99612 

.10540 

.99443 

.12274 

.99244 

.14004 

.99015 

.15730 

.98755 

57 

4 

.08831 

.99609 

.10569 

.99440 

.12302 

.99240 

.14033 

.99011 

.15758 

.98751 

56 

5 

.08860 

.99607 

.10597 

.99437 

.12331 

.99237 

.14061 

.99006 

.15787 

.98746 

55 

6 

.08889 

.99604 

.10626 

.99434 

.12360 

.99233 

.14090 

.99002 

.15816 

.98741 

54 

7 

.08918 

.99602 

.10655 

.99431 

.12389 

.99230 

.14119 

.98998 

.15845 

.98737 

53 

8 

.08947 

.99599 

.10684 

.99428! 

.12418 

.99226 

.14148 

.98994 

.15873 

.98732 

52 

9 

.08976 

.99596 

.10713 

.99424 

.12447 

.99222 

.14177 

.98990 

.15902 

.98728 

51 

10 

.09005 

.99594 

.10742 

.99421 

.12476 

.99219 

.14205 

.98986 

.15931 

.98723 

50 

11 

.09034 

.99591 

.10771 

.99418 

.12504 

.99215 

.14234 

.98982 

.15959 

.98718 

49 

12 

.09063 

.99588 

.10800 

.99415 

.12533 

.99211 

.14263 

.98978 

.15988 

.98714 

48 

13 

.09092 

.99586 

.10829 

.99412 

.12562 

.99208 

.14292 

.98973 

.16017 

.98709 

47 

14 

.09121 

.99583 

.10858 

.99409 

.12591 

.99204 

.14320 

.98969 

.16046 

.98704 

46 

15 

.09150 

.99580 

.10887 

.99406 

.12620 

.99200 

.14349 

.98965 

.1C074 

.98700 

45 

16 

.09179 

.99578 

.10916 

.99402 

.12649 

.99197 

.14370 

.98961 

.16103 

.98695 

44 

17 

.09208 

.99575 

.10945 

.99399 

.12878 

.99193 

.14407 

.98957 

.16132 

.98690 

43 

18 

.09237 

.99572 

.10973 

.99396 

.12706 

.99189 

.14436 

.98953 

.16160 

.98686 

42 

19 

.09288 

.99570 

.11002 

.99393 

.12735 

.99180 

.14464 

.98948 

.16189 

.98681 

41 

20 

.09295 

.99507 

.11031 

.99390 

.12704 

.99182 

.14493 

.98944 

.16218 

.98676 

40 

21 

.09324 

.99564 

.11060 

.99380 

.12793 

.99178 

.14522 

.98940 

.16246 

.98671 

39 

22 

.09353 

.99562 

.11039 

.90333 

• X 

.90175 

.14551 

.98936 

.16275 

.98667 

38 

23 

.09382 

.99559 

.11118 

.90380 

.12351 

.99171 

.14580 

.98931 

.16304 

.98662 

37 

24 

.09411 

.99556 

.11147 

.90377 

.12380 

.99167 

.14608 

.98927 

.16333 

.98657 

36 

25 

.09440 

.99553 

.11176 

.90374 

.12908 

.99163 

.14637 

.98923 

.16361 

.98652 

35 

26 

.09469 

.99551 

.11205 

.99370 

.12937 

.99160 

.14666 

.98919 

.16390 

.98648 

34 

27 

.09498 

.99548 

.11234 

.99367 

.12966 

.99156 

.14695 

.98914 

.16419 

.98643 

33 

28 

.09527 

.99545 

.11263 

.99364 

.12995 

.99152 

.14723 

.98910 

.16447 

.98638 

32 

29 

.09556 

.99542 

.11291 

.99300 

.13024 

.99143 

.14752 

.98906 

.16476 

.98633 

31 

30 

.09585 

.99540 

.11320 

.99357 

.13053 

.99144 

.14781 

.98902 

.16505 

.98629 

30 

31 

.09614 

.99537 

.11349 

.99354 

.13081 

.99141 

.14810 

.98897 

.16533 

.98624 

29 

82 

.09642 

.99534 

.11370 

.90351 

.13110 

.90137 

.14838 

.98893 

.16562 

.98619 

28 

33 

.09671 

.99531 

.11407 

.99347 

.13139 

.99133 

.14867 

.98889 

.16591 

.98614 

27 

34 

.09700 

.99528 

.11436 

.90344 

.13103 

.99120 

.14096 

.98884 

.16620 

.98609 

26 

35 

.09729 

.99526 

.11465 

.99341 

.13197 

.99125 

.14925 

.98880 

.16648 

.98604 

25 

36 

.09758 

.99523 

.11494 

.99337 

.13220 

.99122 

.14954 

.98876 

.16677 

.98600 

24 

37 

.09787 

.99520 

.11523 

.99334 

.13254 

.99118 

.14982 

.98871 

.16706 

.98595 

23 

38 

.09816 

.99517 

.11552 

.99331 

.13283 

.99114 

.15011 

.98867 

.16734 

.98590 

22 

39 

.09845 

.99514 

.11530 

.99327 

.13312 

.99110 

.15040 

.98863 

.16763 

.98585 

21 

40 

.09874 

.99511 

.11609 

.99324 

.13341 

.99106 

.15069 

.98858 

.16792 

.98580 

20 

41 

.09903 

.99508 

.11638 

.99320 

.13370 

.99102 

.15097 

.98854 

.16820 

.98575 

19 

42 

.09932 

.99506 

.11667 

.99317 

.13300 

.99003 

.15126 

.98849 

.16849 

.98570 

18 

43 

.09961 

.99503 

.11006 

.99314 

.13427 

.99004 

.15155 

.98845 

.16878 

.98565 

17 

44 

.09990 

.99500 

.11725 

.99310 

.13450 

.99091 

.15184 

.98841 

.16906 

.98561 

16 

45 

.10019 

.99497 

.11754 

*99307 

.13485 

.99087 

.15212 

.98836 

.16935 

.98556 

15 

46 

.10048 

.99494 

.11783 

.99303 

.13514 

.99083 

.15241 

.98832 

.16964 

.98551 

14 

47 

.10077 

.99491 

.11012 

.99300 

.13543 

.99079 

.15270 

.98827 

.16992 

.98546 

13 

48 

.10106 

.99488 

.11840 

.99297 

.13572 

.99075 

.15290 

.98823 

.17021 

.98541 

12 

49 

.10135 

.99485 

.11069 

.99293 

.13600 

.99071 

.15327 

.98818 

.17050 

.98536 

11 

50 

.10164 

.99482 

.11898 

.99290 

.13629 

.99067 

.15356 

.98814 

.17078 

.98531 

10 

51 

.10192 

.99479 

.11927 

.99286 

.13658 

.99063 

.15385 

.98809 

.17107 

.98526 

9 

52 

.10221 

.99476 

.11956 

.99283 

.13687 

.99059 

.15414 

.98805 

.17136 

.98521 

8 

53 

.10250 

.99473 

.11985 

.99279 

.13716 

.99055 

.15442 

.98800 

.17164 

.98516 

7 

54 

.10279 

.99470 

.12014 

.99276 

.13744 

.99051 

.15471 

.98796 

.17193 

.98511 

6 

55 

.10308 

.99467 

.12043 

.99272 

.13773 

.99047 

.15500 

.98791 

.17222 

.98506 

5 

56 

.10337 

.99464 

.12071 

.99269 

.13802 

.99043 

.15529 

.98787 

.17250 

.98501 

4 

57 

10366 

.99461 

.12100 

.99265 

.13831 

.99039 

.15557 

.98782 

.17279 

.98496 

3 

58 

.10395 

.99458 

.12129 

.99262 

.13860 

.99035 

.15586 

.98778 

.17308 

.98491 

2 

59 

.10424 

.99455 

.12158 

.99258 

.13889 

.99031 

.15615 

.98773 

.17336 

.98486 

1 

60 

.10453 

.99452 

.12187 

.99255 

.13917 

.99027 

.15643 

.98769 

.17365 

.98481 

0 

/ 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

/ 


84 ° 

83 * 

o 

03 

00 

81 ° 

00 

O 

0 



80 





























































NATURAL SINES AND COSINES, 



O 

o 

rH 

11° 

12° 

O 

CO 

f-H 

14° 



Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

> 

0 

.17365 

.98481 

.19081 

.98163 

.20791 

.97815 

.22495 

.97437 

.24192 

.97030 

60 

1 

.17393 

.98476 

.19109 

.98157 

.20820 

.97809 

.22523 

.97430 

.24220 

.97023 

59 

2 

.17422 

.98471 

.19138 

.98152 

.20848 

.97803 

.22552 

.97424 

.24249 

.97015 

58 

3 

.17451 

.98466 

.19167 

.98146 

.20877 

.97797 

.22580 

.97417 

.24277 

.97008 

57 

4 

.17479 

.98461 

.19195 

.98140 

.20905 

.97791 

.22608 

.97411 

.24305 

.97001 

56 

5 

.17508 

.98455 

.19224 

.98135 

.20933 

.97784 

.22637 

.97404 

.24333 

.96994 

55 

6 

.17537 

.98450 

.19252 

.98129 

.20962 

.97778 

.22665 

<97398 

.24362 

.96987 

54 

7 

.17565 

.98445 

.19281 

.98124 

.20990 

.97772 

.22693 

.97391 

.24390 

.96980 

53 

8 

.17591 

.98440 

.19309 

.98118 

.21019 

.97766 

.22722 

.97384 

.24418 

.96973 

52 

9 

.17623 

.98435 

.19338 

.98112 

.21047 

.97760 

.22750 

.97378 

.24446 

.96966 

51 

10 

.17651 

.98430 

.19366 

.98107 

.21076 

.97754 

.22778 

.97371 

.24474 

.96959 

50 

11 

.17680 

.98425 

.19395 

.98101 

.21104 

.97748 

.22807 

.97365 

.24503 

.96952 

49 

12 

.17708 

.98420 

.19423 

.9S096 

.21132 

.97742 

.22835 

.97358 

.24531 

.96945 

48 

13 

.17737 

.98414 

.19452 

.98090 

.21161 

.97735 

.22863 

.97351 

.24559 

.96937 

47 

14 

.17766 

.98409 

.19481 

.98084 

.21189, 

.97729 

.22892 

.97345 

.24587 

.96930 

46 

15 

.17794 

.98404 

.19509 

.98079 

.21218 

.97723 

.22920 

.97338 

.24615 

.96923 

45 

16 

.17823 

.98399 

.19538 

.98073 

.21246 

.97717 

.22948 

.97331 

.24644 

.96916 

44 

17 

.17852 

.98394 

.19566 

.98067 

.21275 

.97711 

.22977 

.97325 

.24672 

.96909 

43 

18 

.17880 

.98389 

.19595 

.98061 

.21303 

.97705 

.23005 

.97318 

.24700 

.96902 

42 

19 

.17909 

.98383 

.19623 

.98056 

.21331 

.97898! 

.23033 

.97311 

.24728 

.96894 

41 

20 

.17937 

.98378 

.19652 

.98050 

.21360 

.97692 

.23062 

.97304 

.24756 

.96887 

40 

21 

.17966 

.98373 

.19680 

.98044 

.21388 

.97686 

.23090 

.97298 

.24784 

.96880 

39 

22 

.17995 

.98368 

.19709 

.98039 

.21417 

.97680 

.23110 

.97291 

.24813 

.96873 

38 

23 

.18023 

.98362 

.19737 

.98033 

.21445 

.97673 

.23146 

.97284 

.24841 

.96866 

37 

24 

.18052 

.98357 

.19766 

.98027 

.21474 

.97667 

.23175 

.97278 

.24869 

.96858 

36 

25 

.18081 

.98352 

.19794 

.98021 

.21502 

.97661 

.23203 

.97271 

.24897 

.96851 

35 

26 

.18109 

.98347 

.19823 

.98016 

.21530 

.97655 

.23231 

.97264 

.24925 

.96844 

34 

27 

.18138 

.98341 

,19851 

.98010 

.21559 

.97648 

.23260 

.97257 

.24954 

.96837 

33 

28 

.18166 

.98336 

.19880 

.98004 

.21587 

.97642 

.23288 

.97251 

.24982 

.96829 

32 

29 

.18195 

.98331 

.19908 

.97998 

.21016 

.97633 

.23316 

.97244 

.25010 

.96822 

31 

30 

.18224 

.98325 

.19937 

.97992 

.21644 

.97630 

.23345 

.97237 

.25038 

.96815 

30 

31 

.18252 

.98320 

.19965 

.97987 

.21672 

.97623 

.23373 

.97230 

.25066 

.96807 

29 

32 

.18281 

.93315 

.19994 

.97981 

.21701 

.97617 

.23401 

.97223 

.25094 

.96800 

28 

33 

.18309 

.98310 

.20022 

.97975 

.21729 

.97611 

.23429 

.97217 

.25122 

.96793 

27 

34 

.18338 

.98304 

.20051 

.97969 

.21758 

.97604 

.23458 

.97210 

.25151 

.96786 

26 

35 

.18367 

.98299 

.20079 

.97983 

.21786 

.97598 

.23488 

.97203 

.25179 

.96778 

25 

36 

.18395 

.98294 

.20108 

.97958 

.21814 

.97592 

.23514 

.97196 

.25207 

.96771 

24 

37 

.18424 

.98288 

.20136 

.97952 

.21843 

.97585 

.23542 

.97189 

.25235 

.96764 

23 

38 

.18452 

.98283 

.20165 

.97948 

.21871 

.97579 

.23571 

.97182 

.25263 

.96756 

22 

39 

.18481 

98277 

.20193 

.97940 

.21899 

.97573 

.23599 

.97176 

.35291 

96749 

21 

40 

.18509 

.98272 

.20222 

.97934 

.21928 

.97566 

.23627 

.97169 

.25320 

.96742 

20 

41 

.18538 

.98267 

.20250 

.97928 

.21956 

.97560 

.23656 

.97162 

.25348 

.96734 

19 

42 

.18567 

.98261 

.20279 

.97922 

.21985 

.97553 

.23684 

.97155 

.25376 

.96727 

18 

43 

.18595 

.98256 

.20307 

.97916 

.22013 

.97547 

.23712 

.97148 

.25404 

.96719 

17 

44 

.18624 

.98250 

.20336 

.97910 

.22041 

.97541 

.23740 

.97141 

.25432 

.96712 

16 

45 

.18652 

.98245 

.20364 

.97905 

.22070 

.97534 

.23769 

.97134 

.25460 

.96705 

15 

46 

.18681 

.98240 

.20393 

.97899 

.22098 

.97528 

.23797 

.97127 

.25488 

.96697 

14 

47 

.18710 

.98234 

.20421 

.97893 

.22126 

.97521 

.23825 

.97120 

.25516 

.96690 

13 

48 

.18738 

.98229 

.20450 

.97887 

.22155 

.97515 

.23853 

.97113 

.25545 

.96682 

12 

49 

.18767 

.98223 

.20478 

.97881 

.22183 

.97508 

.23882 

.97106 

.25573 

.96675 

11 

50 

.18795 

.98218 

.20507 

.97875 

.22212 

.97502 

.23910 

.97100 

.25601 

.96667 

10 

51 

.18824 

.98212 

.20535 

.97869 

.22240 

.97496 

.23938 

.97093 

.25629 

.96660 

9 

52 

.18852 

.98207 

.20563 

.97863 

.22268 

.97489 

.23966 

.97086 

.25657 

.96653 

8 

53 

.18881 

.98201 

.20592 

.97857 

.22297 

.97483 

.23995 

.97079 

.25685 

.96645 

7 

54 

.18910 

.98196 

.20620 

.97851 

.22325 

.97476 

.24023 

.97072 

.25713 

.96638 

6 

55 

.18938 

.98190 

.20649 

.97845 

.22353 

.97470 

.24051 

.97065 

.25741 

.96630 

5 

56 

.18967 

.98185 

.20677 

.97839 

.22382 

.97463 

.24079 

.97058 

.25769 

.96623 

4 

57 

.18995 

.98179 

.20706 

.97833 

.22410 

.97457 

.24108 

.97051 

.25798 

.96615 

3 

58 

.19024 

.98174 

.20734 

.97827 

.22438 

.97450 

.24136 

.97044 

.25826 

.96608 

2 

59 

.19052 

.98168 

.20763 

.97821 

.22467 

.97444 

.24164 

.97037 

.25854 

.96600 

1 

60 

.19081 

.98163 

.20791 

.97815 

.22495 

.97437 

.24192 

.97030 

.258821 

.96593 

0 

/ 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin j 

Sine 

/ 


79° 

CO 

o 

77° 

76° 

75 

o 



81 , 





























































-NATURAL SINES AND COSINES. 



15° 

16° 

17° 

OO 

o 

19° 


9 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 


"o 

.25882 

.96593 

.27564 

.96126 

.29237 

.95630 

.30902 

.95106 

.32557 

.94552 

60 

1 

.25910 

.96585 

.27592 

.96118 

.29265 

.95622 

.30929 

.95097 

.32584 

.94542 

59 

2 

.25938 

.96578 

.27620 

.96110 

.29293 

.95613 

.30957 

.95088 

.32612 

.94533 

58 

3 

.25966 

.96570 

.27648 

.96102 

.29321 

.95605 

.30985 

.95079 

.32639 

.94523 

57 

4 

.25994 

.96562 

.27676 

.96094 

.29348 

.95596 

.3-1012 

.95070 

.32667 

.94514 

56 

5 

.26022 

.96555 

.27704 

.96086 

.29376 

.95588 

.31040 

.95061 

.32694 

.94504 

55 

6 

.26050 

.96547 

.27731 

.96078 

.29404 

.95579 

.31068 

.95052 

.32722 

.94495 

54 

7 

.26079 

.96540 

.27759 

•96070 

.29432 

.95571 

.31095 

.95043 

j .32749 

.94485 

53 

8 

.26107 

.96532 

.27787 

.96062 

.29460 

.95562 

.31123 

.95033 

.32777 

.94476 

52 

9 

.26135 

.96524 

.27815 

.96054 

.29487 

.95554 

.31151 

.95024 

.32804 

.94466 

51 

10 

.26163 

.96517 

.27843 

.96046 

.29515 

.95545 

.31178 

.95015 

.32832 

.94457 

50 

11 

.26191 

.96509 

.27871 

.96037 

.29543 

.95536 

.31206 

.95006 

.32859 

.94417 

49 

12 

.26219 

.96502 

.27899 

.96029 

.29571 

.95528 

.31233 

.94997 

.32887 

.94438 

48 

13 

.26247 

.96494 

.27927 

.96021 

.29599 

.95519 

.31261 

.94988 

.32914 

.94428 

47 

14 

.26275 

.96483 

.27955 

.96013 

.29626 

.95511 

.31289 

.91979 

.32942 

.94418 

46 

15 

.26303 

.96479 

.27983 

.96005 

.29654 

.95502 

.31316 

.94970 

.32969 

.94409 

45 

16 

.26331 

.96471 

.28011 

.95997 

.29682 

.95493 

.31344 

.94961 

.32997 

.94399 

41 

17 

.26359 

.96463 

.28039 

.95989 

.29710 

.95485 

.31372 

.94952 

.33024 

.94390 

43 

18 

.26387 

.96456 

.28067 

.95981 

.29737 

.95476 

.31399 

.94943 

.33051 

.94380 

42 

19 

.26415 

.96448 

.23095 

.95972 

.29765 

.95467 

.31427 

.94933 

,33079 

.94370 

41 

20 

.26443 

.96440 

.28123 

.95964 

.29793 

.95459 

.31454 

.94924 

.33106 

.94361 

40 

21 

.26471 

.96433 

.28150 

.95956 

.29821 

.95450 

.31482 

.94915 

.33134 

.94351 

39 

22 

.26500 

.96425 

.23173 

.95948 

.29849 

.95441 

.31510 

.94906 

.33161 

.94342 

38 

23 

.26528 

.96417 

.28206 

.95940 

.29876 

.95433 

.31537 

.94897 

.33189 

.94332 

37 

24 

.26556 

.96410 

.2S234 

.95931 

.29904 

.95424 

.31565 

.94888 

.33216 

.94322 

36 

25 

.26584 

.98402 

.28262 

.95923 

.29932 

.95415 

.31593 

.94878 

.33244 

.94313 

35 

26 

.26612 

.96394 

.28290 

.95915 

.29960 

.95407 

.31620 

.948G9 

.33271 

.94303 

34 

27 

.26640 

.96386 

.28318 

.95907 

.29987 

.95398 

.31648 

.94860 

.33298 

.94293 

33 

28 

.26668 

.96379 

.28346 

.95898 

.30015 

.95389 

.31G75 

.94851 

.33326 

.94284 

32 

29 

.26696 

.96371 

.28374 

.95890 

.30043 

.95380 

.31703 

.94342 

.33353 

.94274 

31 

30 

.26724 

.96363 

.28402 

.95882 

.30071 

.95372 

.31730 

.94832 

.33381 

.94264 

30 

31 

.26752 

.96355 

.23429 

.95874 

.30098 

.95363 

.31750 

.94823 

.33108 

.94254 

29 

32 

.26780 

.96347 

.23457 

. 9<joG5 

.30126 

.95354 

.31786 

.94814 

.33436 

.94245 

28 

S3 

.26808 

.96340 

.23485 

.95857 

.30154 

.95345 

.31813 

.94805 

.33463 

.94235 

27 

34 

.26836 

.96332 

.28513 

.95849 

.30182 

.95337 

.31841 

.94795 

.33490 

.94225 

26 

35 

.26864 

.96324 

.28541 

.95841 

.30209 

.95323 

.31868 

.94786 

.33518 

.94215 

25 

36 

.26892 

.96316 

.28569 

.95832 

.30237 

.95319 

.31896 

.94777 

.33545 

.94206 

24 

37 

.26920 

.96308 

.28597 

.95824 

.30265 

.95310 

.31923 

.94763 

.33573 

.94196 

23 

38 

.26948 

.96301 

.28625 

.95816 

.30292 

.95301 

.31951 

.94758 

.33600 

.94186 

22 

39 

.26976 

.98293 

.28652 

.95807 

.30320 

.95203 

.31979 

.94749 

.33627 

.94176 

21 

40 

.27004 

.96285 

.28680 

.95799 

.30348 

.95284 

.32006 

.94740 

.33655 

.94167 

20 

41 

.27032 

.96277 

.28708 

.95791 

.30376 

.95275 

.32034 

.94730 

.33682 

.94157 

19 

42 

.27060 

.96269 

.28736 

.95782 

.30403 

.952G6 

.32061 

.94721 

.33710 

.94147 

18 

43 

.27088 

.96261 

.28764 

.95774 

.30431 

.95257 

.32089 

.94712 

.33737 

.94137 

17 

44 

.27116 

.96253 

.28792 

.95766 

.30459 

.95218 

.82116 

.94702 

.33764 

.94127 

16 

45 

.27144 

.96246 

.28§2Q 

.95757 

.30486 

.95240 

.32144 

.94693 

.33792 

.94118 

15 

46 

.27172 

.96238 

.28847 

.95749 

.30514 

.95231 

.32171 

.94084 

.33819 

.94108 

14 

47 

.27200 

.96230 

.28875 

.95740 

.30542 

.95222 

.32199 

.94674 

.33846 

.94098 

13 

48 

.27228 

.96232 

.28903 

.95732 

.30570 

.95213 

.32227 

.94065 

.33874 

.94088 

12 

49 

.27256 

.96214 

.28931 

.95724 

.30597 

.95204 

.32254 

.94656 

33901 

.94078 

11 

50 

.27284 

.96206 

.28959 

.95715 

.30625 

.95195 

.32282 

.94646 

.33929 

.94068 

10 

51 

.27312 

.96198 

.28987 

.95707 

.30653 

.95186 

.32309 

.94637 

.33956 

.94058 

9 

52 

.27340 

.96190 

.29015 

.95698 

.30680 

.95177 

.32337 

.94627 

.33983 

.94049 

8 

53 

.27368 

.96182 

.29042 

.95690 

.30708 

.95168 

.32364 

.94618 

.34011 

.94039 

7 

54 

.27396 

.96174 

.29070 

.95681 

.30736 

.95159 

.32392 

.94609 

.34038 

.94029 

6 

55 

.27424 

.96166 

.29098 

.95673 

.30763 

.95150 

.32419 

.94599 

.34065 

.94019 

5 

56 

.27452 

.96158 

.29126 

.95664 

.30791 

.95142 

.32447 

.94590 

.34093 

.94009 

4 

57 

.27480 

.96150 

.29154 

.95656 

.30819 

.95133 

.32474 

.93580 

.34120 

.93999 

3 

58 

.27508 

.96142 

.29182 

.95647 

.30846 

.95124 

.32502 

.94571 

.34147 

.93989 

2 

59 

.27536 

.96134 

.29209 

.95639 

.30874 

.95115 

.32529 

.94561 

.34175 

.93979 

1 

60 

.27564 

.96126 

.29237 

.95630 

.30902 

.95106 

.32557 

.94552 

.34202 

.93969 

0 

/ 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

§ 


74° 

CO 

0 

72° 

71 

o 

o 

0 



82 



























































natural sines and cosines. 



o 

© 

©i 

21 ° 

22 ° 

23 ° 

to 

4* 

9 

71 


Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine | 

Cosin 


0 

.34202 

.93969 

.35837 

.93358 

.37461 

.92718 

.39073 

.92050 

.40674 

.91355 

60 

1 

.34229 

.93959 

.35864 

.93348 

.37488 

.92707 

.39100 

.92039 

.40700 

.91343 

59 

2 

.34257 

.93949 

.35891 

.93337 

.37515 

.92697 

.39127 

.92028 

.40727 

.91331 

58 

3 

.34284 

.93939 

.35918 

.93327 

.37542 

.92686 

.39153 

.92016 

.40753 

.91319 

57 

4 

.34311 

.93929 

.35945 

.93316 

.37569 

.92675 

.39180 

.92005 

.40780 

.91307 

56 

5 

.34339 

.93919 

.35973 

.93306 

.37595 

.92664 

.39207 

.91994 

.40806 

.91295 

55 

6 

.34366 

.93909 

.36000 

.93295 

.37622 

.92653 

.39234 

.91982 

.40833 

.91283 

54 

7 

.34393 

.93899 

.36027 

.93285 

.37649 

.92642 

.39260 

.91971 

.40860 

.91272 

53 

8 

.34421 

.93889 

.36054 

.93274 

.37676 

.92631 

.39287 

.91959 

.40886 

.91260 

52 

9 

.34448 

.93879 

.36081 

.93264 

.37703 

.92620 

.39314 

.91948 

.40913 

.91248 

51 

10 

.34475 

.93869 

.36108 

.93253 

.37730 

.92609 

.39341 

.91936 

.40939 

.91236 

50 

11 

.34503 

.93859 

.36135 

.93243 

.37757 

.92598 

.39367 

.91925 

.40966 

.91224 

49 

12 

.34530 

.93849 

.36162 

.93232 

.37784 

.92587 

.39394 

.91914 

.40992 

.91212 

48 

13 

.34557 

.93839 

.36190 

.93222 

.37811 

.92576 

.39421 

.91902 

.41019 

.91200 

47 

14 

.34584 

.93829 

.36217 

.93211 

.37838 

.92565 

.39448 

.91891 

.41045 

.91188 

46 

15 

.34612 

.93819 

.36244 

.93201 

.37865 

.92554 

.39474 

.91879 

.41072 

.91176 

45 

16 

.34639 

.93809 

.36271 

.93190 

.37892 

.92543 

.39501 

.91868 

.41098 

.91164 

44 

17 

.34666 

.93799 

.36298 

.93180 

.37919 

.92532 

.39528 

.91856 

.41125 

.91152 

43 

18 

.34694 

.93789 

.36325 

.93169 

.37946 

.92521 

.39555 

.91845 

.41151 

.91140 

42 

19 

.34721 

.93779 

.36352 

.93159 

.37973 

.92510 

.39581 

.91833 

.41178 

.91128 

41 

20 

.34748 

.93769 

.36379 

.93148 

.37999 

.92499 

.39608 

.91822, 

.41204 

.91116 

40 S 

21 

.34775 

.93759 

.36406 

.93137 

.38026 

.92488 

.39635 

.91810 

.41231 

.91104 

39 

22 

.34803 

.93748 

.36434 

.93127 

.38053 

.92477 

.39661 

.91799 

.41257 

.91092 

38 

23 

.34830 

.93738 

.36461 

.93116 

.38080 

.92466 

.39688 

.917871 

.41284 

.91080 

37 

24 

.34857 

.93728 

.36488 

.93106 

.38107 

.92455 

.39715 

.91775 

.41310 

.91068 

36 

25 

.34884 

.93718 

.36515 

.93095 

.38134 

.92444 

.39741 

.91764, 

.41337 

.91056 

35 

26 

.34912 

.93708 

.36542 

.93084 

.38161 

.92432 

.39768 

.91752 

.41363 

.91044 

34 

27 

.34939 

.93698 

.36569 

.93074 

.38188 

.92421 

.39795 

.91741 

.41390 

.91032 

33 

28 

.34966 

.93688 

.36590 

.93063 

.38215 

.92410 

.39822 

.91729 

.41416 

.91020 

32 

29 

.34993 

.93677 

.36623 

.93052 

.38241 

.92399 

.39848 

.91718 

.41443 

.91008 

31 

j 30 

.35021 

.93667 

.36650 

.93042 

.38268 

.92388 

.39875 

.91706 

.41469 

.90996 

30 

31 

.35048 

.93657 

.36677 

.93031 

.38295 

.92377 

.39902 

.91694 

.41496 

.90984 

29 

32 

.35075 

.93647 

.36704 

.93020 

.38322 

.92366 

.39923 

.91683 

.41522 

.90972 

28 1 

33 

.35102 

.93637 

.36731 

.93010 

.38349 

.92355 

.39955 

.91671 

.41549 

.90960 

27 

34 

.35130 

.93626 

.36758 

.92999 

.38370 

.92343 

.39982 

.91660 

.41575 

.90948 

26 

35 

.35157 

.93616 

.36785 

.92988 

.38403 

.92332 

.40008 

.91648 

.41602 

.90936 

25 

36 

.35184 

.93606 

.36812 

.92978 

.38430 

.92321 

.40035 

.91636 

.41628 

.90924 

24 

37 

.35211 

.93596 

.36839 

.92967 

.38456 

.92310 

.40062 

.91625 

.41655 

.90911 

23 

38 

.35239 

.93585 

.36867 

.92956 

.38483 

.92299 

.40088 

.91613 

.41681 

.90899 

22 

39 

.35266 

.93575 

.36894 

.92945 

.38510 

.92287 

.40115 

.91601 

.41707 

.90887 

21 

40 

.35293 

.93565 

.36921 

.92935 

.38537 

.92276 

.40141 

.91590 

.41734 

.90875 

20 

41 

.35320 

.93555 

.36948 

.92924 

.38564 

.92265 

.40168 

.91578 

.41760 

.90863 

19 

42 

.35347 

.93544 

.36975 

.92913 

.38591 

.92254 

.40195 

.91566 

.41787 

.90851 

18 

43 

.35375 

.93534 

.37002 

.92902 

.38617 

.92243 

.40221 

.91555 

.41813 

.90839 

17 

44 

.35402 

.93524 

.37029 

.92892 

.38644 

.92231 

.40248 

.91543 

.41840 

.90826 

16 

45 

.35429 

.93514 

.37056 

.92881 

.38671 

.92220 

.40275 

.91531 

.41866 

.90814 

15 

46 

.35456 

.93503 

.37083 

.92870 

.38698 

.92209 

.40301 

.91519 

.41892 

.90802 

14 

47 

.35484 

.93493 

.37110 

.92859 

.38725 

.92198 

.40328 

.91508 

.41919 

.90790 

13 

48 

.35511 

.93483 

.37137 

.92849 

.38752 

.92186 

.40355 

.91496 

.41945 

.90778 

12 

49 

.35538 

.93472 

.37164 

.92838 

.38778 

.92175 

.40381 

.91484 

.41972 

.90766 

11 

50 

.35565 

.93462 

.37191 

.92827 

.38805 

.92164 

.40408 

.91472 

.41998 

.90753 

10 

51 

.35592 

.93452 

.37218 

.92816 

.38832 

.92152 

.40434 

.91461 

.42024 

.90741 

9 

52 

.35619 

.93441 

.37245 

.92805 

.38859 

.92141 

.40461 

.91449 

.42051 

.90729 

8 

53 

.35647 

.93431 

.37272 

.92794 

.38886 

.92130 

.40488 

.91437 

.42077 

.90717 

7 

54 

.35674 

.93420 

.37299 

.92784 

.38912 

.92119 

.40514 

.91425 

.42104 

.90704 

6 

55 

.35701 

.93410 

.37326 

.92773 

.38939 

.92107 

.40541 

.91414 

.42130 

.90692 

5 

56 

.35728 

.93400 

.37353 

.92762 

.38966 

.92096 

.40567 

.91402 

.42156 

.90680 

4 

57 

.35755 

.93389 

.37380 

.92751 

.38993 

.92085 

.40594 

.91390 

.42183 

.90668 

3 

58 

.35782 

.93379 

.37407 

.92740 

.39020 

.92073 

.40621 

.91378 

.42209 

.90655 

2 

59 

.35810 

.93368 

.37434 

.92729 

.39046 

.92062 

.40647 

.91366 

.42235 

.90643 

1 

60 

.35837 

.93358 

.37461 

.92718 

.39073 

.92050 

.40674 

.91355 

.42262 

.90631 

0 

/ 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

/ 


69 ° 

0 

00 

© 

67 ° 

66 ° 

65 * 

" 3 


83 































































NATURAL SINES AND COSINES, 



25 ° 

| 26 ° 

27 ° 

o 

QO 

29 ° 

~ 7 | 


Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosir 


0 

.4226$ 

.90631 

.4383- 

- .89879 

.4539$ 

.89101 

.46941 

.88295 

.48481 

.8746$ 

no 

1 

.42288 

.90618 

.43861 

.89867 

. 4542E 

.89087 

.46972 

i .88281 

.48506 

.87448 

59 

2 

.42315 

.90606 

.43881 

.89854 

. 4545] 

.89074 

.4699$ 

.88267 

.48532 

.87434 

58 

3 

.42341 

.90594 

.43911 

.89841 

.45477 

.89061 

.47024 

.88254 

.48557 

.8742C 

57 

4 

.42367 

.90582 

.4394$ 

.89828 

.45503 

.89048 

.4705C 

.88240 

.48583 

.87406 

56 

5 

.42394 

.90569 

.4396!: 

.89816 

.45529 

.89035 

.47076 

.88226 

.48608 

.87391 

55 

6 

.42420 

.90557 

.43994 

.89803 

.45554 

.89021 

.47101 

.88213 

.48634 

.87377 

54 

7 

.42446 

.90545 

.4402C 

.89790 

.45580 

.89008 

.47127 

.88199 

.48659 

.87363 

53 

8 

.42473 

.90532 

.44046 

.89777 

.45606 

.88995 

.47153 

.88185 

.48684 

.87349 

52 

9 

.42499 

.90520 

.44072 

.89764 

.45632 

.88981 

.47178 

.88172 

.48710 

.87335 

51 

10 

.42525 

.90507 

.44098 

.89752 

.45658 

.88968 

.47204 

.88158 

.48735 

.87321 

50 

11 

.42552 

.90495 

.44124 

.89739 

.45684 

.88955 

.47229 

.88144 

.48761 

.87306 

49 

12 

.42578 

.90483 

.44151 

.89726 

.45710 

.88942 

.47255 

.88130 

.48786 

.87292 

48 

13 

.42604 

.90470 

.44177 

.89713 

.45736 

.88928 

.47281 

.88117 

.48811 

.87278 

47 

14 

.42631 

.90458 

.44203 

.89700 

.45762 

.88915 

.47306 

.88103 

.48837 

.87264 

46 

15 

.42657 

.90446 

.44229 

.89687 

.45787 

.88902 

.47332 

.88089 

.48862 

.87250 

45 

16 

.42683 

.90433 

.44255 

.89674 

.45813 

.88888 

.47358 

.88075 

.48888 

.87235 

44 

17 

.42709 

.90421 

.44281 

.89662 

.45839 

.88875 

.47383 

.88062 

.48913 

.87221 

43 

18 

.42736 

.90408 

.44307 

.89649 

.45865 

.88862 

.47409 

.88048 

.48938 

.87207 

42 

19 

.42762 

.90396 

.44333 

.89636 

.45891 

.88848 

.47434 

.88034 

.48964 

.87193 

41 

20 

.42788 

.90383 

.44359 

.89623 

.45917 

.88835 

.47460 

.88020 

.48989 

.87178 

40 

21 

.42815 

.90371 

.44385 

.89610 

.45942 

.88822 

.47486 

.88006 

.49014 

.87164 

39 

22 

.42841 

.90358 

.44411 

.89597 

.45968 

.88803 

.47511 

.87993 

.49040 

.87150 

38 

23 

.42867 

.90346 

.44437 

.89584 

.45994 

.88795 

.47537 

.87979 

.49065 

.87136 

37 

24 

.42894 

.90334 

.44464 

.89571 

.46020 

.88782 

.47562 

.87965 

.49090 

.87121 

36 

25 

.42920 

.90321 

.44490 

.89558 

.46046 

.88768 

.47588 

.87951 

.49116 

.87107 

35 

26 

.42946 

.90309 

.44516 

.89545 

.46072 

.88755 

.47614 

.87937 

.49141 

.87093 

34 

27 

.42972 

.90296 

.44542 

.89532 

.46097 

.88741 

.47639 

.87923 

.49166 

.87079 

33 

28 

.42999 

.90284 

.44568 

.89519 

.46123 

.88728 

.47665 

.87909 

.49192 

.87064 

32 

29 

.43025 

.93271 

.44594 

.89506 

.40140 

.88715 

.47690 

.87896 

.49217 

.87050 

31 

30 

.43051 

.90259 

.44620 

.89493 

.46175 

.88701 

.47716 

.87882 

.49242 

.87036 

30 

31 

.43077 

.90246 

.44646 

.89480 

.46201 

.88688 

.47741 

.87868 

.49268 

.87021 

29 

32 

.43104 

.90233 

.44672 

.89407 

.46223 

.83674 

.47767 

.87854 

.49293 

.87007 

28 

33 

.43130 

.90221 

.44693 

.89454 

.46252 

.88681 

.47793 

.87840 

.49318 

.86993 

27 

34 

.43156 

.90208 

.44724 

.89441 

.46278 

.88647 

.47818 

.87826 

.49344 

.86978 

26 

35 

.43182 

.90196 

.44750 

.89428 

.46304 

.88634 

.47844 

.87812 

.49369 

.86964 

25 

36 

.43209 

.90183 

.44776 

.89415 

.46330 

.88620 

.47869 

.87708 

.49394 

.86949 

24 

' 37 

.43235 

.90171 

.44802 

.89402 

.46355 

.88607 

.47895 

.87784 

.49419 

.86935 

23 

38 

.43261 

.90158 

.44828 

.89389 

.46381 

.88593 

.47920 

.87770 

.49445 

.86921 

22 

39 

.43287 

.90146 

.44854 

.89376 

.46407 

.88580 

.47946 

.87756 

.49470 

.86906 

21 

40 

.43313 

.90133 

.44880 

.89303 

.46433 

.88566 

.47971 

.87743 

.49495 

.86892 

20 

i 41 

.43340 

.90120 

.44906 

.89350 

.46458 

.88553 

.47997 

.87729 

.49521 

.86878 

19 

42 

.43366 

.90108 

.44932 

.89337 

.46484 

.83539 

.48022 

.87715 

.49546 

.86863 

18 

43 

.43392 

.90095 

.44958 

.89324 

.46510 

.8852G 

.48048 

.87701 

.49571 

.86849 

17 

44 

.43418 

.90082 

.44984 

.89311 

.46536 

.88512 

.48073 

.87687 

.49596 

.86834 

16 

45 

.43445 

.90070 

.45010 

.89298 

.46501 

.88499 

.48099 

.87673 

.49622 

.86820 

15 

•46 

.43471 

.90057 

.45036 

.89285 

.46587 

.88485 

.48124 

.87659 

.49647 

.86805 

14 

47 

.43497 

.90045 

.45062 

.89272 

.40613 

.88472 

.48150 

.87645 

.49672 

.86791 

13 

48 

.43523 

.90032 

.45088 

.89259 

.46639 

.88458 

.48175 

.87631 

.49697 

.86777 

12 

49 

.43549 

.90019 

.45114 

.89245 

.46664 

.88445 

.48201 

.87617 

.49723 

.86762 

11 

50 

.43575 

.90007 

.45140 

.89232 

.46690 

.88431 

.48226 

.87603 

.49748 

.86748 

10 

51 

.43602 

.89994 

.45166 

.89219 

.46716 

.88417 

.48252 

.87589 

.49773 

.86733 

9 

52 

.43628 

.89981 

.45192 

.89206 

.46742 

.88404 

.48277 

.87575 

.49798 

.86719 

8 

53 

.43654 

.89968 

.45218 

.89193 

.46767 

.88390 

.48303 

.87561 

.49824 

.86704 

7 

54 

.43680 

.89956 

.45243 

.89180 

.46793 

.88377 

.48328 

.87546 

.49849 

.86690 

6 

55 

.43706 

.89943 

.45269 

.89167 

.46819 

.88363 

.48354 

.87532 

.49874 

86675 

5 

56 

. 437 33 

.89930 

.45295 

.89153 

.46844 

.88349 

.48379 

.87518 

.49899 

86661 

4 

57 

.43759 

.89918 

.45321 

.89140 

.46870 

.88336 

.48405 

87504 

.49924 

86646 

3 

58 

.43785 

.89905 

.45347 

.89127 

.46896 

.88322 

.48430 

87490 

.49950 

86632 

2 

59 

.43811 

.89892 

.45373 

.89114 

.46921 

.88308 

.48456 

87476 

.49975 

86617 

1 

60 

.43837 

.89879 

.45399 

.89101 

.46947 

.88295 

.48481 

87462 

.50000 

86603 

0 

/ 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine j 


i ■ 

64 ° 

63 

o 

62 

O 

61 ' 

> 

60 ' 




64 





































































































NATURAL SINES AND COSINES. 



30 

* 1 

31 

° i 

32 

O 

33 ° 

CO 

0 

1 



Sine 

Cosin 

Sine 

Cosin ' 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 


0 

.50000 

.86603 

.51504 

.85717 

.52992 

.84805 

.54464 

.83867 

.55919 

.82904 

60 

1 

.50025 

.86588 

.51529 

.85702 

.53017 

.84789 

.54488 

.83851 

.55943 

.82887 

59 

2 

.50050 

.86573 

.51554 

.85687 

.53041 

.84774 

.54513 

.83835 

.55968 

.82871 

58 

3 

.50076 

.86559 

.51579 

.85672 

.53066 

.84759 

.54537 

.83819 

.55992 

.82855 

57 

4 

.50101 

.86544 

.51604 

.85657 

.53091 

.84743 

.54561 

.83804 

.56016 

.82839 

56 

5 

.50126 

.86530 

.51628 

.85642 

.53115 

.84728 

.54586 

.83788 

.56040 

.82822 

55 

6 

.50151 

.86515 

.51653 

.85627 

.53140 

.84712 

.54610 

.83772 

.56064 

.82806 

54 

7 

.50176 

.86501 

.51678 

.85612 

.53164 

.84697 

.54635 

.83756 

.56088 

.82790 

53 

8 

.50201 

.86486 

.51703 

.85597 

.53189 

.84681 

.54659 

.83740 

.56112 

.82773 

52 

9 

.50227 

.86471 

.51728 

.85582 

.53214 

.84666 

.54683 

.83724 

.56136 

.82757 

51 

10 

.50252 

.86457 

.51753 

.85567 

.53238 

.84650 

.54708 

.83708 

.56160 

.82741 

50 

11 

.50277 

.86442 

.51778 

.85551 

.53263 

.84635 

.54732 

.83692 

.56184 

.82724 

49 

12 

.50302 

.86427 

.51803 

.85536 

.53288 

.84619 

.54756 

.83676 

.56208 

.82708 

48 

13 

.50327 

.86413 

.51828 

.85521 

.53312 

.84604 

.54781 

.83660 

.56232 

.82692 

47 

14 

.50352 

.86398 

.51852 

.85506 

.53337 

.84588 

.54805 

.83645 

.56256 

.82675 

46 

15 

.50377 

.86384 

.51877 

.85491 

.53361 

.84573 

.54829 

.83629 

.56280 

.82659 

45 

16 

.50403 

.86369 

.51902 

.85476 

.53386 

.84557 

.54854 

.83613 

.56305 

.82643 

44 

17 

.50428 

.86354 

.51927 

.85461 

.53411 

.84542 

.54878 

.83597 

.56329 

.82626 

43 

18 

.50453 

.86340 

.51952 

.85446 

.53435 

.84526 

.54902 

.83581 

.56353 

.82610 

42 

19 

.50478 

.86325 

.51977 

.85431 

.53460 

.84511 

.54927 

.83565 

.56377 

.82593 

41 

20 

.50503 

.86310 

.52002 

.85416 

.53484 

.84495 

.54951 

.83549 

.56401 

.82577 

40 

21 

.50528 

.86295 

.52026 

.85401 

.53509 

.84480 

.54975 

.83533 

.56425 

.82561 

39 

22 

.50553 

.86281 

.52051 

.85385 

.53534 

.84464 

.54999 

.83517 

.56449 

.82544 

38 

23 

.50578 

.86266 

.52076 

.85370 

.53558 

.84448 

.55024 

.83501 

.56473 

.82528 

37 

24 

.50603 

.86251 

.52101 

.85355 

.53583 

.84433 

.55048 

.83485 

.56497 

.82511 

36 

25 

.50628 

.86237 

.52126 

.85340 

.53607 

.84417 

.55072 

.83469 

.56521 

.82495 

35 

26 

.50654 

.86222 

.52151 

.85325 

.53632 

.84402 

.55097 

.83453 

.56545 

.82478 

34 

27 

.50879 

.86207 

.52175 

.85310 

.53656 

.84386 

.55121 

.83437 

.56569 

.82462 

33 

28 

.50704 

.86192 

.52200 

.85294 

.53681 

.84370 

.55145 

.83421 

.56593 

.82446 

32 

29 

.50729 

.86178 

.52225 

.85279 

.53705 

.84355 

.55169 

.83405 

.56617 

.82429 

31 

30 

.50754 

.86163 

.52250 

.85264 

.53730 

.84339 

.55194 

.83389 

.56641 

.82413 

30 

31 

.50779 

.86148 

.52275 

.85249 

.53754 

.84324 

.55218 

.83373 

.56665 

.82396 

29 

32 

.50804 

.86133 

.52299 

.85234 

.53779 

.84308 

.55242 

.83356 

.56689 

.82380 

28 

33 

.50829 

.86119 

.52324 

.85218 

.53804 

.84292 

.55266 

.83340 

.56713 

.82363 

27 

34 

.50854 

.86104 

.52349 

.85203 

.53828 

.84277 

.55291 

.83324 

.56736 

.82347 

26 

35 

.50879 

.86089 

.52374 

.85188 

.53853 

.84261 

.55315 

.83308 

.56760 

.82330 

25 

36 

.50904 

.86074 

.52399 

.85173 

.53877 

.84245 

.55339 

.83292 

.56784 

.82314 

24 

37 

.50929 

.86059 

.52423 

.85157 

.53902 

.84230 

.55363 

.83276 

.56808 

.82297 

23 

38 

.50954 

.86045 

.52448 

.85142 

.53926 

.84214 

.55388 

.83260 

.56832 

.82281 

22 

39 

.50979 

.86030 

.52473 

.85127 

.53951 

.84198 

.55412 

.83244 

.56856 

.82264 

21 

40 

.51004 

.86015 

.52498 

.85112 

.53975 

.84182 

.55436 

.83228 

.56880 

.82248 

20 

41 

.51029 

.86000 

.52522 

.85096 

.54000 

.84167 

.55460 

.83212 

.56904 

.82231 

19 

42 

.51054 

.85985 

.52547 

.85081 

.54024 

.84151 

.55484 

.83195 

.56928 

.82214 

18 

43 

.51079 

.85970 

.52572 

.85066 

.54049 

.84135 

.55509 

.83179 

.56952 

.82198 

17 

44 

.51104 

.85956 

.52597 

.85051 

.54073 

.84120 

.55533 

.83163 

.56976 

.82181 

16 

45 

.51129 

.85941 

.52621 

.85035 

.54097 

.84104 

.55557 

.83147 

.57000 

.82165 

15 

46 

.51154 

.85926 

.52646 

.85020 

.54122 

.84088 

.55581 

.83131 

.57024 

.82148 

14 

47 

.51179 

.85911 

.52671 

.85005 

.54146 

.84072 

.55605 

.83115 

.57047 

.82132 

13 

48 

.51204 

.85896 

.52696 

.84989 

.54171 

.84057 

.55630 

.83098 

.57071 

.82115 

12 

49 

.51229 

.85881 

.52720 

.84974 

.54195 

.84041 

.55654 

.83082 

.57095 

.82098 

11 

50 

.51254 

.85866 

.52745 

.84959 

.54220 

.84025 

.55678 

.83066 

.57119 

.82082 

10 

51 

.51279 

.85851 

.52770 

.84943 

.54244 

.84009 

.55702 

.83050 

.57143 

.82065 

9 

52 

.51304 

.85836 

.52794 

.84928 

.54269 

.83994 

.55726 

.83034 

.57167 

.82048 

8 

53 

.51329 

.85821 

.52819 

.84913 

.54293 

.83978 

.55750 

.83017 

.57191 

.82032 

7 

54 

.51354 

.85806 

.52844 

.84897 

.54317 

.83962 

.55775 

.83001 

.57215 

.82015 

6 

55 

.51379 

.85792 

.52869 

.84882 

.54342 

.83946 

.55799 

.82985 

.57238 

.81999 

5 

56 

.51404 

.85777 

.52893 

.84866 

.54366 

.83930 

.55823 

.82969 

.57262 

.81982 

4 

57 

.51429 

.85762 

.52918 

.84851 

.54391 

.83915 

.55847 

.82953 

.57286 

.81965 

3 

58 

.51454 

.85747 

.52943 

.84836 

.54415 

.83899 

.55871 

.82936 

.57310 

.81949 

2 

59 

.51479 

.85732 

.52967 

.84820 

.54440 

.83863 

.55895 

.82920 

.57334 

.81932 

1 

60 

.51504 

.85717 

.52992 

.84805 

.54464 

.83867 

.55919 

.82904 

.57358 

.81915 

0 

f 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

f 


1 69 ° 

58 ° 

57 ° 

56 ° 

55 ° 



85 








































































NATURAL SINES AND COSINES, 



CO 

c* 

o 

36 ° 

CO 

o 

38 ° 

39 ° 



Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosir 

/ 

0 

.57358 

.81915 

.58779 

.80902 

.60182 

.79864 

.61566 

.78801 

.6293$ 

i .77711 

60 

1 

.57381 

.81899 

.58802 

.80885 

.60205 

.79846 

.61589 

.78783 

.62951 

> . 7769( 

j 59 

2 

.57405 

.81882 

.58826 

.80867 

.60228 

.79829 

.61612 

.78765 

,6297i 

. 77678 

58 

3 

.57429 

.81865 

.58849 

.80850 

.60251 

.79811 

.61635 

.78747 

. 6300( 

) .77661 

57 

4 

.57453 

.81848 

.58873 

.80833 

.60274 

.79793 

.61658 

.78729 

.6302$ 

.77641 

56 

5 

.57477 

.81832 

.58896 

.80816 

.60298 

.79776 

.61681 

.78711 

. 63041 

. 7762c 

55 

6 

.57501 

.81815 

.58920 

.80799 

.60321 

.79758 

.61704 

.78694 

. 63066 

.77601 

54 

7 

.57524 

.81798 

.58943 

.80782 

.60344 

.79741 

.61726 

.78676 

.6309C 

.77586 

53 

8 

.57548 

.81782 

.58967 

.80765 

.60367 

.79723 

.61749 

.78658 

.63113 

.77568 

52 

9 

.57572 

.81765 

.58990 

.80748 

.60390 

.79706 

.61772 

.78640 

.63135 

.7755C 

51 

10 

.57596 

.81748 

.59014 

.80730 

.60414 

.79688 

.61795 

.78622 

.63158 

.77531 

50 

11 

.57619 

.81731 

.59037 

.80713 

.60437 

.79671 

.61818 

.78604 

.63180 

.77513 

49 

12 

.57643 

.81714 

.59061 

.80696 

.60460 

.79653 

.61841 

.78586 

.63203 

.77494 

48 

13 

.57667 

.81698 

.59084 

.80679 

.60483 

.79635 

.61864 

.78568 

.63225 

.77476 

47 

14 

.57691 

.81681 

.59108 

.80662 

.60506 

.79618 

.61887 

.78550 

.63248 

.77458 

46 

15 

.57715 

.81664 

.59131 

.80644 

.60529 

.79600 

.61909 

.78532 

.63271 

.77439 

45 

16 

.57738 

.81647 

.59154 

.80627 

.60553 

.79583 

.61932 

.78514 

.63293 

.77421 

44 

17 

.57762 

.81631 

.59178 

.80610 

.60576 

.79565 

.61955 

.78496 

.63316 

.77402 

43 

18 

.57786 

.81614 

.59201 

.80593 

.60599 

.79547 

.61978 

.78478 

.63338 

.77384 

42 

19 

.57810 

.81597 

.59225 

.80576 

.60622 

.79530 

.62001 

.78460 

.63361 

.77366 

41 

20 

.57833 

.81580 

.59248 

.80558 

.60645 

.79512 

.62024 

.78442 

.63383 

.77347 

40 

21 

.57857 

.81563 

.59272 

.80541 

.60668 

.79494 

.62046 

.78424 

.63406 

.77329 

39 

22 

.57881 

.81546 

.59295 

.80524 

.60691 

.79477 

.62069 

.78405 

.63428 

.77310 

38 

23 

.57904 

.81530 

.59318 

.80507 

.60714 

.79459 

.62092 

.78387 

.63451 

.77292 

37 

24 

.57928 

.81513 

.59342 

.80489 

.60738 

.79441 

.02115 

.78369 

.63473 

.77273 

36 

25 

.57952 

.81496 

.59365 

.80472 

.60761 

.79424 

.62138 

.78351 

.63496 

.77255 

35 

26 

.57976 

.81479 

.59389 

.80455 

.60784 

.79406 

.62160 

.78333 

.63518 

.77236 

34 

27 

.57999 

.81462 

.59412 

.80438 

.60807 

.79388 

.62183 

.78315 

.63540 

.77218 

33 

28 

.58023 

.81445 

.59436 

.80420 

.60830 

.79371 

.62206 

.78297 

.63563 

.77199 

32 

29 

.58047 

.81428 

.59459 

.80403 

60853 

.79353 

.62229 

.78279 

.63585 

.77181 

31 

30 

.58070 

.81412 

.59482 

.80386 

.60876 

.79335 

.62251 

.78261 

.63608 

.77162 

30 

31 

.58094 

.81395 

.59506 

.80368 

.60899 

.79318 

.62274 

.78243 

.63630 

.77144 

29 

32 

.58118 

.81378 

.59529 

.80351 

.60922 

.79300 

.62297 

.78225 

.63653 

.77125 

28 

33 

.58141 

.81361 

.59552 

.80334 

.60945 

.79282 

.62320 

.78206 

.63675 

.77107 

27 

34 

.58165 

.81344 

.59576 

.80316 

.60968 

.79264 

.62342 

.78188 

„63698 

.77088 

26 

35 

.58189 

.81327 

.59599 

.80299 

.60991 

.79247 

.62365 

.78170 

.63720 

.77070 

25 

36 

.58212 

.81310 

.59622 

.80282 

.61015 

.79229 

.62388 

.78152 

.63742 

.77051 

24 

37 

.58236 

.81293 

.59646 

.80264 

.61038 

.79211 

.62411 

.78134 

.63765 

.77033 

23 

38 

.58260 

.81276 

.59669 

.80247 

.61061 

.79193 

.62433 

.78116 

.63787 

.77014 

22 

39 

.58283 

.81259 

.59693 

.80230 

.61084 

.79176 

.62456 

.78098 

.63810 

.76996 

21 

40 

.58307 

.81242 

.59716 

.80212 

.61107 

.79158 

.62479 

.78079 

.63832 

.76977 

20 

41 

.58330 

.81225 

.59739 

.80195 

.61130 

.79140 

.62502 

.78061 

.63854 

.76959 

19 

42 

.58354 

.81208 

.59763 

.80178 

.61153 

.79122 

.62524 

.78043 

.63877 

.76940 

18 

43 

.58378 

.81191 

.59786 

.80160 

.61176 

.79105 

.62547 

.78025 

.63899 

.76921 

17 

44 

.58401 

.81174 

.59809 

.80143 

.61199 

.79087 

.62570 

.78007 

.63922 

.76903 

16 

45 

.58425 

.81157 

.59832 

.80125 

.61222 

.79069 

.62592 

.77988 

.63944 

.76884 

15 

46 

.58449 

.81140 

.59856 

.80108 

.61245 

.79051 

.62615 

.77970 

.63966 

.76866 

14 

47 

.58472 

.81123 

.59879 

.80091 

.61268 

.79033 

.62638 

.77952 

.63989 

.76847 

13 

48 

.58496 

.81106 

.59902 

.80073 

.61291 

.79016 

.62660 

.77934 

.64011 

.76828 

12 

49 

.58519 

.81089 

.59926 

.80056 

.61314 

.78998 

.62683 

.77916 

.64033 

.76810 

11 

50 

.58543 

.81072 

.59949 

.80038 

.61337 

.78980 

.62706 

.77897 

.64056 

.76791 

10 

51 

.58567 

.81055 

.59972 

.80021 

.61360 

.78962 

.62728 

.77879 

.64078 

76772 

9 

52 

.58590 

.81038 

.59995 

.80003 

.61383 

.78944 

.62751 

.77861 

.64100 

.76754 

8 

53 

.58614 

.81021 

.60019 

.79986 

.61406 

.78926 

.62774 

.77843 

.64123 

76735 

7 

54 

.58637 

.81004 

.60042 

.79968 

.61429 

.78908 

.62796 

77824 

.64145 

76717 

6 

55 

.58661 

.80987 

.60065 

.79951 

.61451 

.78891 

.62819 

77806 

.64167 

76698 

5 

56 

.58684 

.80970 

.60089 

.79934 

.61474 

.78873 

.62842 

77788 

.64190 

76679 

4 

57 

.58708 

80953 

.60112 

.79916 

.61497 

.78855 

.62864 

77769 

.64212 

76661 

3 

58 

.58731 

.80936 

.60135 

.79899 

.61520 

.78837 

.62887 

77751 

.64234 

76642 

2 

59 

.58755 

.80919 

.60158 

.79881 

.61543 

78819 

.62909 

77733 

.64256 

76623 

1 

60 

.58779 

.80902 

.60182 

.79864 

.61566 

78801 

.62932 

77715 

.64279 

76604 

0 

/ 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 



64 ° 1 

53 

O 

52 

O j 

51 

» 

50 

0 



.86 


















































































NATURAL SINES AND COSINES. 



o 

O 

41 ° 

42 ° 

CO 

o 

l 

44 ° 



Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

/ 

0 

.64279 

.76604 

.65606 

.75471 

.66913 

.74314 

.68200 

.73135 

.69466 

.71934 

60 

1 

.64301 

.76586 

.65628 

.75452 

.66935 

.74295 

.68221 

.73116 

.69487 

.71914 

59 

2 

.64323 

.76567 

.65650 

.75433 

.66956 

.74276 

.68242 

.73096 

.69508 

.71894 

58 

3 

.64346 

.76548 

.65672 

.75414 

.66978 

.74256 

.68264 

.73076 

.69529 

.71873 

57 

4 

.64368 

.76530 

.65694 

.75395 

.66999 

.74237 

.68285 

.73056 

.69549 

.71853 56 

5 

.64390 

.76511 

.65716 

.75375 

.67021 

.74217 

.68306 

.73036 

.69570 

.71833 

55 

6 

.64412 

.76492 

.65738 

.75356 

.67043 

.74198 

.68327 

.73016 

.69591 

.71813 

54 

7 

.64435 

.76473 

.65759 

.75337 

.67064 

.74178 

.68349 

.72996 

.69612 

.71792 

53 

8 

.64457 

.76455 

.65781 

.75318 

.67086 

.74159 

.68370 

.72976 

.69633 

.71772 

. 52 

9 

.64479 

.76436 

.65803 

.75299 

.67107 

.74139 

.68391 

.72957 

.69654 

.71752 51 

10 

.64501 

.76417 

.65825 

.75280 

.67129 

.74120 

.68412 

.72937 

.69675 

.71732 

50 

11 

.64524 

.76398 

.65847 

.75261 

.67151 

.74100 

.68434 

.72917 

.69696 

.71711 

49 

12 

.64546 

.76380 

.65869 

.75241 

.67172 

.74080 

.68455 

.72897 

.69717 

.71691 

48 

13 

.64568 

.76331 

.65891 

.75222 

.67194 

.74061 

.68476 

.72877 

.69737 

.71671 

47 

14 

.64590 

.76342 

.65913 

.75203 

.67215 

.74041 

.68497 

.72857 

.69758 

.71650 

46 

15 

.64612 

.76323 

.65935 

.75184 

.67237 

.74022 

.68518 

.72837 

.69779 

.71630 

45 

16 

.64635 

.76304 

.65956 

.75165 

.67258 

.74002 

.68539 

.72817 

.69800 

.71610 

44 

17 

.64657 

.76286 

.65978 

.75146 

.67280 

.73983 

.68561 

.72797 

.69821 

.71590 

43 

18 

.64679 

.76267 

.66000 

.75128 

.67301 

.73963 

.68582 

.72777 

.69842 

.71569 

42 

19 

.64701 

.76248 

.66022 

.75107 

.67323 

.73944 

.68603 

.72757 

.69862 

.71549 

41 

20 

.64723 

.76229 

.66044 

.75088 

.67344 

.73924 

.68624 

.72737 

.69883 

.71529 

40 

21 

.64746 

.76210 

.66066 

.75089 

.67366 

.73904 

.68645 

.72717 

.69904 

.71508 

39 

22 

.64768 

.76192 

.66088 

.75050 

.67387 

.73885 

.68666 

.72697 

.69925 

.71488 

38 

23 

.64790 

.76173 

.68109 

.75030 

.67409 

.73865 

.68688 

.72677 

.69946 

.71468 

37 

24 

.64812 

.76154 

.66131 

.75011 

.67430 

.73846 

.68709 

.72657 

.69966 

.71447 

36 

25 

.64834 

.76135 

.66153 

.74992 

.67452 

.73826 

.68730 

.72637 

.69987 

.71427 

35 

26 

.64856 

.76116 

.63175 

.74973 

.67473 

.73803 

.68751 

.72617 

.70008 

.71407 

34 

27 

.64878 

.76097 

.66197 

.74953 

.67495 

.73787 

.68772 

.72597 

.70029 

.71386 

33 

28 

.64901 

.76078 

.68218 

.74934 

.67516 

.73767 

.68793 

.72577 

.70049 

.71366 

32 

29 

.64923 

.76059 

.68240 

.74915 

.67538 

.73747 

.68814 

.72557 

.70070 

.71345 

31 

30 

.64945 

.76041 

.66262 

.74896 

.67559 

.73728 

.68835 

.72537 

.70091 

.71325 

30 

31 

.64967 

.76022 

.66284 

.74876 

.67580 

.73708 

.68857 

.72517 

.70112 

.71305 

29 

32 

.64989 

.76003 

.66308 

.74857 

.67602 

.73688 

.68878 

.72497 

70132 

.71284 

28 

33 

.65011 

.75984 

.66327 

.74838 

.67623 

.73669 

.63899 

.72477 

.70153 

.71264 

27 

34 

.65033 

.75965 

.66349 

.74818 

.67645 

.73649 

.68920 

.72457 

.70174 

.71243 

26 

35 

.65055 

.75946 

.66371 

.74799 

.67666 

.73629 

.68941 

.72437 

.70195 

.71223 

25 

36 

.65077 

.75927 

.66393 

.74780 

.67688 

.73610 

.68962 

.72417 

.70215 

.71203 

24 

37 

.65100 

.75908 

.66414 

.74760 

.67709 

.73590 

.68983 

.72397 

.70236 

.71182 

23 

38 

.65122 

.75889 

.68438 

.74741 

.67730 

.73570 

.69004 

.72377 

.70257 

.71162 

22 

39 

.65144 

.75870 

.68458 

.74722 

.67752 

.73551 

.69025 

.72357 

.70277 

.71141 

21 

40 

.65166 

.75851 

.68480 

.74703 

.67773 

.73531 

.69046 

.72337 

.70298 

.71121 

20 

41 

.65188 

.75832 

.66501 

.74683 

.67795 

.73511 

.69067 

.72317 

.70319 

.71100 

19 

42 

.65210 

.75813 

.68523 

.74664 

.67816 

.73491 

.69088 

.72297 

.70339 

.71080 

18 

43 

.65232 

.75794 

.68545 

.74644 

.67837 

.73472 

.69109 

.72277 

.70360 

.71059 

17 

44 

.65254 

.75775 

.66568 

.74625 

.67859 

.73452 

.69130 

.72257 

.70381 

.71039 

16 

45 

.65276 

.75756, 

.66588 

.74606 

.67880 

.73432 

.69151 

.72236 

.70401 

.71019 

15 

46 

.65298 

.75738 

.66610 

.74586 

.67901 

.73413 

.69172 

.72216 

.70422 

.70998 

14 

47 

.65320 

.75719 

.66632 

.74567 

.67923 

.73393 

.69193 

.72196 

.70443 

.70978 

13 

48 

.65342 

.757001 

.66653 

.74548 

.67944 

.73373 

.69214 

.72176 

.70463 

.70957 

12 

49 

.65364 

.75680 

.66675 

.74528 

.67965 

.73353 

.69235 

.72156 

.70484 

.70937 

11 

50 

.65386 

.75661 

.66697 

.74509 

.67987 

.73333 

.69256 

.72136 

.70505 

.70916 

10 

51 

.65408 

.75642 

.66718 

.74489 

.68008 

.73314 

.69277 

.72116 

.70525 

.70896 

9 

52 

.65430 

.75623 

.66740 

.74470 

.68029 

.73294 

.69298 

.72095 

.70546 

.70875 

8 

53 

.65452 

.75604 

.66762 

.74451 

.68051 

.73274 

.69319 

.72075 

.70567 

.70855 

7 

54 

.65474 

.75585 

.66783 

.74431 

.68072 

.73254 

.69340 

.72055 

.70587 

.70834 

6 

55 

.65496 

.75566 

.66805 

.74412 

.68093 

.73234 

.69361 

.72035 

.70608 

.70813 

5 

56 

.65518 

.75547 

.66827 

.74392 

.68115 

.73215 

.69382 

.72015 

.70628 

.70793 

4 

57 

.65540 

.75528 

.66848 

.74373 

.68136 

.73195 

.69403 

.71995 

.70649 

.70772 

3 

58 

.65562 

.75509 

.66870 

.74353 

.68157 

.73175 

.69424 

.71974 

.70670 

.70752 

2 

59 

.65584 

.75490 

.66891 

.74334 

.68179 

.73155 

.69445 

.71954 

.70690 

.70731 

1 

60 

.65606 

.75471 

.66913 

.74314 

.68200 

.73135 

.69466 

.71934 

.70711 

.70711 

0 

/ 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

/ 


49 ° 

o 

CO 

47 ° 

46 ° 

45 ° 



87 



























































NATURAL TANGENTS AND COTANGENTS. 


r 

0 ° 

1 

O 

2 

5° 

a 

!° 



Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 


~o 

.00000 

Infinite. 

.01746 

57.2900 

.03492 

28.6363 

.05241 

19.0811 

60 

4 

1 

.00029 

3437.75 

.01775 

56.3506 

.03521 

28.3994 

.05270 

18.9755 

59 

2 

.00058 

1718.87 

.01804 

55.4415 

.03550 

28.1664 

.05299 

18.8711 

58 

3 

.00087 

1145.92 

.01833 

54.5613 

.03579 

27.9372 

.05328 

18.7678 

57 

4 

.00116 

859.436 

.01862 

53.7086 

.03609 

27.7117 

.05357 

18.6656 

56 

5 

.00145 

687.549 

.01891 

52.8821 

.03638 

27.4899 

.05387 

18.5645 

55 

6 

.00175 

572.957 

.01920 

52.0807 

.03667 

27.2715 

.05416 

18.4645 

54 

7 

.00204 

491.106 

.01949 

51.3032 

.03696 

27.0566 

.05445 

18.3655 

53 

8 

.00233 

429.718 

.01978 

50.5485 

.03725 

26.8450 

.05474 

18.2677 

52 

9 

.00262 

381.971 

.02007 

49.8157 

.03754 

26.6367 

.05503 

18.1708 

51 

10 

.00291 

343.774 

.02036 

49.1039 

.03783 

26.4316 

.05533 

18.0750 

50 

11 

.00320 

312.521 

.02066 

48.4121 

.03812 

26.2296 

.05562 

17.9802 

49 

12 

.00349 

286.478 

.02095 

47.7395 

.03842 

26.0307 

.05591 

17.8863 

48 

13 

.00378 

264.441 

.02124 

47.0853 

.03871 

25.8348 

.05620 

17.7934 

47 

14 

.00407 

245.552 

.02153 

46.4489 

.03900 

25.6418 

.05649 

17.7015 

46 

15 

.00436 

229.182 

.02182 

45.8294 

.03929 

25.4517 

.05678 

17.6106 

45 

16 

.00465 

214.858 

.02211 

45.2261 

.03958 

25.2644 

.05708 

17.5205 

44 

17 

.00495 

202.219 

.02240 

44.6386 

.03987 

25.0798 

.05737 

17.4314 

43 

18 

.00524 

190.984 

.02269 

44.0661 

.04016 

24.8978 

.05766 

17.3432 

42 

19 

.00553 

180.932 

.02298 

43.5081 

.04046 

24.7185 

.05795 

17.2558. 

41 

20 

.00582 

171.885 

.02328 

42.9641 

.04075 

24.5418 

.05824 

17.1693 

40 

21 

.00611 

163.700 

.02357 

42.4335 

.04104 

24.3675 

.05854 

17.0837 

39 

22 

.00640 

156.259 

.02386 

41.9158 

.04133 

24.1957 

.05083 

16.9990 

38 

23 

.00669 

149.465 

.02415 

41.4106 

.04162 

24.0263 

.05912 

16.9150 

37 

24 

.00698 

143.237 

.02444 

40.9174 

.04191 

23.8593 

.05941 

16.8319 

36 

25 

.00727 

137.507 

.02473 

40.4358 

.04220 

23.6945 

.05970 

16.7496 

35 

26 

.00756 

132.219 

.02502 

39.9655 

.04250 

23.5321 

.05999 

16.6681 

34 

27 

.00785 

127.321 

.02531 

39.5059 

.04279 

23.3718 

.06029 

16.5874 

33 

28 

.00815 

122.774 

.02560 

39.0568 

.04303 

23.2137 

.06058 

16.5075 

32 

29 

.00844 

118.540 

.02589 

38.6177 

.04337 

23.0577 

.06087 

16.4283 

31 

30 

.00873 

114.589 

.02619 

38.1885 

.04366 

22.9038 

.06116 

16.3499 

30 

31 

.00902 

110.892 

.02648 

37.7686 

.04395 

22.7519 

.06145 

16.2722 

29 

82 

.00931 

107.426 

.02677 

37.3579 

.04424 

22.6020 

.06175 

16.1952 

28 

33 

.00960 

104.171 

.02706 

36.9560 

.04454 

22.4541 

.06204 

16.1190 

27 

34 

.00989 

101.107 

.02735 

36.5627 

.04483 

22.3081 

.06233 

16.0435 

26 

35 

.01018 

98.2179 

.02764 

36.1776 

.04512 

22.1640 

.06262 

15.9687 

25 

36 

.01047 

95.4895 

.02793 

35.8006 

.04541 

22.0217 

.06291 

15.8945 

24 

37 

.01076 

92.9085 

.02822 

35.4313 

.04570 

21.8813 

.06321 

15.8211 

23 

38 

.01105 

90.4633 

.02851 

35.0695 

.04599 

21.7426 

.06350 

15.7483 

22 

89 

.01135 

88.1436 

.02881 

34.7151 

.04628 

21.6056 

.06379 

15.6762 

21 

4(9 

.01164 

85.9398 

.02910 

34.3678 

.04658 

21.4704 

.06408 

15.6048 

20 

41 

.01193 

83.8435 

.02939 

34.0273 

.04687 

21.3369 

.06437 

15.5340 

19 

42 

.01222 

81.8470 

.02968 

33.6935 

.04716 

21.2049 

.06467 

15.4638 

18 

4.$ 

.01251 

79.9434 

.02997 

33.3662 

.04745 

21.0747 

.06496 

15.3943 

17 

44 

.01280 

78.1263 

.03026 

33.0452 

.04774 

20.9460 

.06525 

15.3254 

16 

45 

.01309 

76.3900 

.03055 

32.7303 

.04803 

20.8188 

.06554 

15.2571 

15 

46 

.01338 

74.7292 

.03084 

32.4213 

.04833 

20.6932 

.06584 

15.1893 

14 

47 

.01367 

73.1390 

.03114 

32.1181 

.04862 

20.5691 

.06613 

15.1222 

13 

48 

.01396 

71.6151 

.03143 

31.8205 

.04891 

20.4465 

.06642 

15.0557 

12 

49 

.01425 

70.1533 

.03172 

31.5284 

.04920 

20.3253 

.06671 

14.9898 

11 

50 

.01455 

68.7501 

.03201 

31.2416 

.04949 

20.2056 

.06700 

14.9244 

10 

51 

.01484 

67.4019 

.03230 

30.9599 

.04978 

20.0872 

.06730 

14.8596 

9 

52 

.01513 

66.1055 

.03259 

30.6833 

.05007 

19.9702 

.06759 

14.7954 

8 

53 

.01542 

64.8580 

.03288 

30.4116 

.05037 

19.8546 

.06788 

14.7317 

7 

54 

.01571 

63.6567 

.03317 

30.1446 

.05066 

19.7403 

.06817 

14.6685 

6 

55 

.01600 

62.4992 

.03346 

29.8823 

.05095 

19.6273 

.06847 

14.6059 

5 

56 

.01629 

61.3829 

.03376 

29.6245 

.05124 

19.5156 

.06876 

14.5438 

4 

57 

.01658 

60.3058 

.03405 

29.3711 

.05153 

19.4051 

.06905 

14.4823 

3 

58 

.01687 

59.2659 

.03434 

29.1220 

.05182 

19.2959 

.06934 

14.4212 

2 

5& 

.01716 

58.2612 

.03463 

28.8771 

.05212 

19.1879 

.06963 

14.3607 

1 

60 

.01746 

57.2900 

.03492 

28.6363 

.05241 

19.0811 

.06993 

14.3007 

0 

/ 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

/ 


89 ° 

00 

oo 

o 

00 

o 

86 ° 



88 














































NATURAL TANGENTS AND COTANGENTS. 



4 

° 

5 

)° 

6 

° 

7 

O 



Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

/ 

0 

.06993 

14.3007 

.08749 

11.4301 

.10510 

9.51436 

.12278 

8.14435 

60 

1 

.07022 

14.2411 

.08778 

11.3919 

.10540 

9.48781 

.12308 

8.12481 

59 

2 

.07051 

14.1821 

.08807 

11.3540 

.10569 

9.46141 

.12338 

8.10536 

58 

3 

.07080 

14.1235 

.08837 

11.3163 

.10599 

9.43515 

.12367 

8.08600 

57 

4 

.07110 

14.0655 

.08866 

11.2789 

.10628 

9.40904 

.12397 

8.06674 

56 

5 

.07139 

14.0079 

.08895 

11.2417 

.10657 

9.38307 

.12426 

8.04756 

55 

6 

.07168 

13.9507 

.08925 

11.2048 

.10687 

9.35724 

.12456 

8.02848 

54 

7 

.07197 

13.8940 

.08954 

11.1681 

.10716 

9.33155 

.12485 

8.00948 

53 

8 

.07227 

13.8378 

.08983 

11.1316 

.10746 

9.30599 

.12515 

7.99058 

52 

9 

.07256 

13.7821 

.09013 

11.0954 

.10775 

9.28058 

.12544 

7.97176 

51 

10 

.07285 

13.7267 

.09042 

11.0594 

.10805 

9.25530 

.12574 

7.95302 

50 

11 

.07314 

13.6719 

.09071 

11.0237 

.10834 

9.23016 

.12603 

7.93438 

49 

12 

.07344 

13.6174 

.09101 

10.9882 

.10863 

9.20516 

.12633 

7.91582 

48 

13 

.07373 

13.5634 

.09130 

10.9529 

.10893 

9.18028 

.12662 

7.89734 

47 

14 

.07402 

13.5098 

.09159 

10.9178 

.10922 

9.15554 

.12692 

7.87895 

46 

15 

.07431 

13.4566 

.09189 

10.8829 

.10952 

9.13093 

.12722 

7.86064 

45 

16 

.07461 

13.4039 

.09218 

10.8483 

.10981 

9.10646 

.12751 

7.84242 

44 

17 

.07490 

13.3515 

.09247 

10.8139 

.11011 

9.08211 

.12781 

7.82428 

43 

18 

.07519 

13.2996 

.09277 

10.7797 

.11040 

9.C57C9 

.12810 

7.80622 

42 

19 

.07548 

13.2480 

.09306 

10.7457 

.11070 

9.C3379 

.12840 

7.78825 

41 

20 

.07578 

13.1969 

.09335 

10.7119 

.11099 

9.00983 

.12869 

7.77035 

40 

21 

.07607 

13.1461 

.09365 

10.6783 

.11128 

8.98598 

.12899 

7.75254 

39 

22 

.07636 

13.0958 

.09394 

10.6450 

.11158 

8.CG227 

.12929 

7.73480 

38 

23 

.07665 

13.0458 

.09423 

10.6118 

.11187 

8 93867 

.12958 

7.71715 

37 

24 

.07695 

12.9962 

.09453 

10.5789 

.11217 

8.91520 

.12988 

7.69957 

36 

25 

»07724 

12.9469 

.09482 

10.5462 

.11246 

8.89185 

.13017 

7.68208 

35 

26 

.07753 

12.8981 

.09511 

10.5136 

.11276 

8.86862 

.13047 

7.66466 

34 

27 

.07782 

12.8496 

.09541 

10.4813 

.11305 

8.84551 

.13076 

7.64732 

33 

28 

.07812 

12.8014 

.09570 

10.4491 

.11335 

8.82252 

.13106 

7.63005 

32 

29 

.07841 

12.7536 

.09600 

10.4172 

.11364 

8.79964 

.13136 

7.61287 

31 

30 

.07870 

12.7062 

.09629 

10.3854 

.11394 

8.77689 

.13165 

7.59575 

30 

31 

.07899 

12.6591 

.09658 

10.3538 

.11423 

8.75425 

.13195 

7.57872 

29 

32 

.07929 

12.6124 

.09688 

10.3224 

.11452 

8.73172 

.13224 

7.56176 

28 

33 

.07958 

12.5660 

.09717 

10.2913 

.11482 

8.70931 

.13254 

7.54487 

27 

34 

.07987 

12.5199 

.09746 

10.2602 

.11511 

8.68701 

.13284 

7.52806 

26 

35 

.08017 

12.4742 

.09776 

10.2294 

.11541 

8.66482 

.13313 

7.51132 

25 

36 

.08046 

12.4288 

.09805 

10.1988 

.11570 

8.64275 

.13343 

7.49465 

24 

37 

.08075 

12.3838 

.09834 

10.1683 

.11600 

8.62078 

.13372 

7.47806 

23 

38 

.08104 

12.3390 

.09864 

10.1381 

.11629 

8.59893 

.13402 

7.46154 

22 

39 

.08134 

12.2946 

.09893 

10.1080 

.11659 

8.57718 

.13432 

7.44509 

21 

40 

.08163 

12.2505 

.09923 

10.0780 

.11688 

8.55555 

.13461 

7.42871 

20 

41 

.08192 

12.2067 

.09952 

10.0483 

.11718 

8.53402 

.13491 

7.41240 

19 

42 

.08221 

12.1632 

.09981 

10.0187 

.11747 

8.51259 

.13521 

7.39616 

18 

43 

.08251 

12.1201 

.icon 

9.98931 

.11777 

8.49128 

.13550 

7.37999 

17 

44 

.08280 

12.0772 

.10040 

9.96007 

.11803 

8.47007 

.13580 

7.36389 

16 

45 

.08309 

12.0346 

.10069 

9.93101 

.11836 

8.44896 

.13609 

7.34786 

15 

46 

.08339 

11.9923 

.10099 

9.90211 

.11865 

8.42795 

.13639 

7.33190 

14 

47 

.08368 

11.9504 

.10128 

9.87338 

.11895 

8.40705 

.13669 

7.31600 

13 

48 

.08397 

11.9087 

.10158 

9.84482 

.11924 

8.38625 

.13698 

7.30018 

12 

49 

.08427 

11.8673 

.10187 

9.81641 

.11954 

8.36555 

.13728 

7.28442 

11 

50 

.08456 

11.8262 

.10216 

9.78817 

.11983 

8.34496 

.13758 

7.26873 

10 

51 

.08485 

11.7853 

.10246 

8.76009 

.12013 

8.32446 

.13787 

7.25310 

9 

52 

.08514 

11.7448 

.10275 

9.73217 

.12042 

8.30406 

.13817 

7.23754 

8 

53 

.08544 

11.7045 

.10305 

9.70441 

.12072 

8.28376 

.13846 

7.22204 

7 

54 

.08573 

11.6645 

.10334 

9.67680 

.12101 

8.26355 

.13876 

7.20661 

6 

55 

.08602 

11.6248 

.10363 

9.64935 

.12131 

8.24345 

.13906 

7.19125 

5 

56 

.08632 

11.5853 

.10393 

9.62205 

.12160 

8.22344 

.13935 

7.17594 

4 

57 

.08661 

11.5461 

.10422 

9.59490 

.12190 

8.20352 

.13965 

7.16071 

3 

58 

.08690 

11.5072 

.10452 

9.56791 

.12219 

8.18370 

.13995 

7.14553 

2 

59 

.08720 

11.4685 

.10481 

9.54106 

.12249 

8.16398 

.14024 

7.13042 

1 

60 

.08749 

11.4301 

.10510 

9.51436 

.12278 

8.14435 

.14054 

7.11537 

0 

' 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

/ 


0 

lO 

CO 

o 

CO 

83° 

82° 


















































NATURAL TANGENTS AND COTANGENTS, 



go 

9 ° 

10 ° 

| 11 ° 



Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

" / 

0 

.14054 

7.11537 

.15838 

6.31375 

.17633 

5.67128 

.19438 

5.14455 

60 

1 

.14084 

7.10038 

.15868 

6.30189 

.17663 

5.66465 

.19468 

5.13658 

59 

2 

.14113 

7.08546 

.15898 

6.29007 

.17693 

5.65205 

.19498 

5.12862 

58 

3 

.14143 

7.07059 

.15928 

6.27829 

.17723 

5.64248 

.19529 

5.12069 

57 

4 

.14173 

7.05579 

.15958 

6.26655 

.17753 

5.63295 

.19559 

5.11279 

56 

5 

.14202 

7.04105 

.15988 

6.25486 

.17783 

5.62344 

.19589 

5.10490 

55 

6 

.14232 

7.02637 

.16017 

6.24321 

.17813 

5.61397 

.19619 

5.09704 

54 

7 

.14262 

6.91174 

.16047 

6.23160 

.17843 

5.60452 

.19649 

5.08921 

53 

8 

.14291 

6.99718 

.16077 

6.22003 

.17873 

5.59511 

.19680 

5.08139 

52 

9 

.14321 

6.98268 

.16107 

6.20851 

.17903 

5.58573 

.19710 

5.07360 

51 

10 

.14351 

6.96823 

.16137 

6.19703 

.17933 

5.57638 

.19740 

5.06584 

50 

11 

.14381 

6.95385 

.16167 

6.18559 

.17963 

5.56706 

.19770 

5.05809 

49 

12 

.14410 

6.93952 

.16196 

6.17419 

.17993 

5.55777 

.19801 

5.05037 

48 

13 

.14440 

6.92525 

.16226 

6.16283 

.18023 

5.54851 

.19831 

5.04267 

47 

14 

.14470 

6.91104 

.16256 

6.15151 

.18053 

5.53927 

.19861 

5.03499 

46 

15 

.14499 

6.89688 

.16286 

6.14023 

.18083 

5.53007 

.19891 

5.02734 

45 

16 

.14529 

6.88278 

.16316 

6.12899 

.18113 

5.52090 

.19921 

5.01971 

44 

17 

.14559 

6.86874 

.16346 

6.11779 

.18143 

5.51176 

.19952 

5.01210 

43 

18 

.14588 

6.85475 

.16376 

6.10354 

.18173 

5.50264 

.19982 

5.00451 

42 

19 

.14618 

6.84082 

.16405 

6.09552 

.18203 

5.49356 

.20012 

4.99695 

41 

20 

.14648 

6.82694 

.16435 

6.08444 

.18233 

5.48451 

.20042 

4.98940 

40 

21 

.14678 

6.81812 

.16465 

6.07340 

.18263 

5.47548 

.20073 

4.98188 

39 

22 

.14707 

6.79936 

.16495 

6.06240 

.18293 

5.46648 

.20103 

4.97438 

33 

23 

.14737 

6.78564 

.16525 

6.05143 

.18323 

5.45751 

.20133 

4.96690 

37 

24 

.14767 

6.77199 

.16555 

6.04051 

.18353 

5.44857 

.20164 

4.95945 

36 

25 

.14796 

6.75838 

.16585 

6.02962 

.18334 

5.43966 

.20194 

4.95201 

35 

26 

.14826 

6.74483 

.16615 

6.01878 

.18414 

5.43077 

.20224 

4.94460 

34 

27 

.14856 

6.73133 

.16645 

6.00797 

.18444 

5.42192 

.20254 

4.93721 

33 

28 

.14886 

6.71789 

.16674 

5.99720 

.18474 

5.41309 

(.20285 

4.92984 

32 

29 

.14915 

6.70450 

.16704 

5.98646 

.18504 

5.40429 

.20315 

4.92249 

31 

30 

.14945 

6.69116 

.10734 

5.97576 

.18534 

5.39552 

.20345 

4.91516 

30 

91 

.14975 

6.67787 

.16764 

5.96510 

.18564 

5.38677 

.20376 

4.90785 

29 

32 

.15005 

6.66463 

.16794 

5.95448 

.18594 

5.37805 

.20406 

4.90056 

28 

33 

.15034 

6.65144 

.16824 

5.94390 

.18624 

5.36936 

.20436 

4.89330 

27 

34 

.15064 

6.63831 

.16854 

5.93335 

.18654 

5.36070 

.20466 

4.88605 

26 

35 

.15094 

6.62523 

.16884 

5.92283 

.18684 

5.35206 

.20497 

4.878S2 

25 

36 

.15124 

6.61219 

.16914 

5.91236 

.18714 

5.34345 

.20527 

4.87162 

24 

?7 

.15153 

6.59921 

.16944 

5.90191 

.18745 

5.33487 

.20557 

4.86444 

23 

38 

.15183 

6.5S627 

.16974 

5.89151 

.18775 

5.32631 

.20588 

4.85727 

22 

39 

.15213 

6.57339 

.17004 

5.88114 

.18805 

5.31778 

.20618 

4.85013 

21 

40 

.15243 

6.56055 

.17033 

5.87080 

.18835 

5.30928 

.20648 

4.84300 

20 

41 

.15272 

6.54777 

.17063 

5.86051 

.18865 

5.30080 

.20679 

4.83590 

19 

42 

.15302 

6.53503 

.17093 

5.85024 

.18895 

5.29235 

.20709 

4.82882 

18 

43 

.15332 

6.52234 

.17123 

5.84001 

.18925 

5.28393 

.20739 

4.82175 

17 

44 

.15362 

6.50970 

.17153 

5.82982 

.18955 

5.27553 

.20770 

4.81471 

16 

45 

.15391 

6.49710 

.17183 

5.81966 

.18986 

5.26715 

.20800 

4.80769 

15 

16 

.15421 

6.48456 

.17213 

5.80953 

.19016 

5.258S0 

.20830 

4.80068 

14 

47 

.15451 

6.47206 

.17243 

5.79944 

.19046 

5.25048 

.20861 

4.79370 

13 

48 

.15481 

6.45961 

.17273 

5.78938 

.19076 

5.24218 

.20891 

4.78673 

12 

49 

.15511 

6.44720 

.17303 

5.77936 

.19106 

5.23391 

.20921 

4.77978 

11 

50 

.15540 

6.43484 

.17333 

5.76937 

.19136 

5.22566 

.20952 

4.77286 

10 

irl 

.15570 

6.42253 

.17363 

5.75941 

.19165 

5.21744 

.20982 

4.76595 

9 

52 

.15600 

6.41026 

.17393 

5.74949 

.19197 

5.20925 

.21018 

4.75906 

8 

53 

.15630 

6.39804 

.17423 

5.73960 

.19227 

5.20107 

.21043 

4.75219 

7 

54 

.15660 

6.38587 

.17453 

5.72974 

.19257 

5.19293 

.21073 

4.74534 

6 

55 

.15689 

6.37374 

.17483 

5.71992 

.19287 

5.18480 

.21104 

4.73851 

5 

56 

.15719 

6.36165 

.17513 

5.71013 

.19317 

5.17671 

.21134 

4.73170 

4 

57 

.15749 

6.34961 

.17543 

5.70037 

.19347 

5.16863 

.21164 

4.72490 

3 

58 

.15779 

6.33761 

.17573 

5.69064 

.19378 

5.16058 

.21195 

4.71813 

2 

59 

.15809 

6.32566 

.17603 

5.68094 

.19408 

5.15256 

.21225 

4.71137 

1 

60 

.15838 

6.31375 

.17633 

5.67128 

.19438 

5.14455 

.21256 

4.70463 

0 

0 

Cotang 

Tang 

Co tang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

/ 

r - 

81 ° 

80 ° 

79 ° 

78 ° 

J 


90 











































































NATURAL TANGENTS AND COTANGENTS. 



12° 

13° 

14° 

15° 



Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

/ 

0 

.21256 

4.70463 

.23087 

4.33148 

.24933 

4.01078 

.26795 

3.73205 

60 

1 

.21286 

4.69791 

.23117 

4.32573 

.24964 

4.00582 

.26826 

3.72771 

59 

2 

.21316 

4.69121 

.23148 

4.32001 

.24995 

4.00086 

.26857 

3.72338 

58 

3 

.21347 

4.68452 

.23179 

4.31430 

.25026 

3.99592 

.26888 

3.71907 

57 

4 

.21377 

4.67786 

.23209 

4.30860 

.25056 

3.99099 

.28920 

3.71476 

56 

5 

.21408 

4.67121 

.23240 

4.30291 

.25087 

3.98607 

.26951 

3.71046 

55 

6 

.21438 

4.66458 

.23271 

4.29724 

.25118 

3.98117 

.26982 

3.70616 

54 

7 

.21469 

4.65797 

.23301 

4.29159 

.25149 

3.97627 

.27013 

3.70188 

53 

8 

.21499 

4.65138 

.23332 

4.28595 

.25180 

3.97139 

.27044 

3.69761 

52 

9 

.21529 

4.64480 

.23363 

4.28032 

.25211 

3.96651 

.27076 

3.69335 

51 

10 

.21560 

4.63825 

.23393 

4.27471 

.25242 

3.96165 

.27107 

3.68909 

50 

11 

.21590 

4.63171 

.23424 

4.26911 

.25273 

3.95680 

.27138 

3.68485 

49 

12 

.21621 

4.62518 

.23455 

4.26352 

.25304 

3.95196 

.27109 

3.68061 

48 

13 

.21651 

4.61868 

.23485 

4.25795 

.25335 

3.94713 

.27201 

3.67638 

47 

14 

.21682 

4.61219 

.23516 

4.25239 

.25366 

3.94232 

.27232 

3.67217 

46 

15 

.21712 

4.60572 

.23547 

4.24685 

.25397 

3.93751 

.27263 

3.66796 

45 

16 

.21743 

4.59927 

.23578 

4.24132 

.25428 

3.93271 

.27294 

3.66376 

44 

17 

.21773 

4.59283 

.23608 

4.23580 

.25459 

3.92793 

.27326 

3.65957 

43 

18 

.21804 

4.58641 

.23639 

4.23030 

.25490 

3.92316 

.27357 

3.65538 

42 

19 

.21834 

4.58001 

.23070 

4.22481 

.25521 

3.91839 

.27388 

3.65121 

41 

20 

.21864 

4.57363 

.23700 

4.21933 

.25552 

3.91364 

.27419 

3.64705 

40 

21 

.21895 

4.56726 

.23731 

4.21387 

.25583 

3.90890 

.27451 

3.64289 

39 

22 

.21925 

4.56091 

.237C2 

4.20842 

.25614 

3.90417 

.27482 

3.63874 

38 

23 

.21956 

4.55458 

.23793 

4.20298 

.25645 

3.89945 

.27513 

3.63461 

37 

24 

.21986 

4.54826 

.23823 

4.19756 

.25676 

3.89474 

.27545 

3.63048 

36 

25 

.22017 

4.54196 

.23854 

4.19215 

.25707 

3.89004 

.27576 

3.62636 

35 

26 

.22047 

4.53568 

.238C5 

4.18675 

.25738 

3.88536 

.27607 

3.62224 

34 

27 

.22078 

4.52941 

.23916 

4.13137 

.25769 

3.88008 

.27638 

3.61814 

33 

28 

.22108 

4.52316 

.23946 

4.17600 

.25800 

3.87601 

.27070 

3.61405 

32 

29 

.22139 

4.51693 

.23977 

4.17064 

.25831 

3.87136 

.27701 

3.60996 

31 

30 

.22169 

4.51071 

.24008 

4.16530 

.25862 

3.86671 

.27732 

3 00588 

30 

31 

.22200 

4.50451 

.24039 

4.15997 

.25893 

3.86208 

.27764 

3.60181 

29 

32 

.22231 

4.49832 

.24069 

4.15465 

.25924 

3.85745 

.27795 

3.59775 

28 

33 

.22261 

4.49215 

.24100 

4.14934 

.25955 

3.85284 

.27825 

3.59370 

27 

34 

.22292 

4.48600 

.24131 

4.14405 

.25986 

3.84824 

.27858 

3.58966 

26 

35 

.22322 

4.47986 

.24162 

4.13877 

.26017 

3.84364 

.27889 

3.58562 

25 

36 

.22353 

4.47374 

.24193 

4.13350 

.26048 

3.83906 

.27921 

3.58160 

24 

37 

.22383 

4.46764 

.24223 

4.12825 

.26079 

3.83449 

.27952 

3.57758 

23 

38 

.22414 

4.46155 

.24254 

4.12301 

.26110 

3.82992 

.27983 

3.57357 

22 

39 

.22444 

4.45548 

.24285 

4.11778 

.26141 

3.82537 

.28015 

3.56957 

21 

40 

.22475 

4.44942 

.24316 

4.11258 

.26172 

3.82083 

.28046 

3.56557 

20 

41 

.22505 

4.44338 

.24347 

4.10736 

.26203 

3.81630 

.28077 

3.56159 

19 

42 

.22536 

4.43735 

.24377 

4.10216 

.26235 

3.81177 

.28109 

3.55761 

18 

43 

.22567 

4.43134 

.24408 

4.09699 

.26266 

3.80726 

.28140 

3.55364 

17 

44 

.22597 

4.42534 

.24439 

4.09182 

.26297 

3.80276 

.28172 

3.54968 

16 

45 

.22628 

4.41936 

.24470 

4.08666 

.26328 

3.79827 

.28203 

3.54573 

15 

46 

.22658 

4.41340 

.24501 

4.08152 

.26359 

3.79378 

.28234 

3.54179 

14 

47 

.22689 

4.40745 

.24532 

4.07639 

.26390 

3.78931 

.28266 

3.53785 

13 

48 

.22719 

4.40152 

.24562 

4.07127 

.26421 

3.78485 

.28297 

3.53393 

12 

49 

.22750 

4.39560 

.24593 

4.06616 

.26452 

3.78040 

.28329 

3.53001 

11 

50 

.22781 

4.38969 

.24624 

4.06107 

.26483 

3.77595 

.28360 

3.52609 

10 

51 

.22811 

4.38381 

.24655 

4.05599 

.26515 

3.77152 

.28391 

3.52219 

9 

52 

.22842 

4.37793 

.24686 

4.05092 

.26546 

3.76709 

.28423 

3.51829 

8 

53 

.22872 

4.37207 

.24717 

4.04586 

.26577 

3.76268 

.28454 

3.51441 

7 

54 

.22903 

4.36623 

.24747 

4.04081 

.26608 

3.75828 

.28486 

3.51053 

6 

55 

.22934 

4.36040 

.24778 

4.03578 

.26639 

3.75388 

.28517 

3.50666 

5 

56 

.22964 

4.35459 

.24809 

4.03076 

.26670 

3.74950 

.28549 

3.50279 

4 

57 

.22995 

4.34879 

.24840 

4.02574 

.26701 

3.74512 

.28580 

3.49894 

3 

58 

.23026 

4.34300 

.24871 

4.02074 

.26733 

3.74075 

.28612 

3.49509 

2 

59 

.23056 

4.33723 

.24902 

4.01576 

.26764 

3.73640 

.28643 

3.49125 

1 

60 

.23087 

4.33148 

.24933 

4.01078 

.26795 

3.73205 

.28675 

3.48741 

0 


Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

/ 


77° 

76° 

75° 

74° 



91 





































.'Natural tangents and cotangents, 



16° 

17° 

1 18° 

19° 



Tang 

Cotang 

Tang 

Cotang 

! Tang 

Cotang 

Tang 

Cotang 

/ 

0 

.28675 

3.48741 

.30573 

3.27085 

.32492 

3.07768 

.34433 

2.90421 

“60 

1 

.28706 

3.48359 

.30605 

3.26745 

.32524 

3.07464 

.34465 

2.90147 

59 

2 

.28738 

3.47977 

.30637 

3.26406 

.32556 

3.07160 

.34498 

2.89873 

58 

3 

.28769 

3.47596 

.30669 

3.26067 

.32588 

3.06857 

.34530 

2.89600 

57 

4 

.28800 

3.47216 

.30700 

3.25729 

.32621 

3.06554 

.34563 

2.89327 

56 

5 

.28832 

3.46837 

.30732 

3.25392 

.32653 

3.06252 

.34596 

2.89055 

55 

6 

.28864 

3.46458 

.30764 

3.25055 

.32685 

3.05950 

.34628 

2.88783 

54 

7 

.28895 

3.46080 

.30796 

3.24719 

.32717 

3.05649 

.34661 

2.88511 

53 

8 

.28927 

3.45703 

.30828 

3.24383 

.32749 

3.05349 

.34693 

2.88240 

52 

9 

.28958 

3.45327 

.30860 

3.24049 

.32782 

3.05049 

.34726 

2.87970 

51 

10 

.28990 

3.44951 

.30891 

3.23714 

.32814 

3.04749 

.34758 

2.87700 

50 

11 

.29021 

3.44576 

.30923 

3.23381 

.32846 

3.04450 

.34791 

2.87430 

49 

12 

.29053 

3.44202 

.30955 

3.23048 

.32878 

3.04152 

.34824 

2.87161 

48 

13 

.29084 

3.43829 

.30987 

3.22715 

.32911 

3.03854 

.34856 

2.86892 

47 

14 

.29116 

3.43456 

.31019 

3.22384 

.32943 

3.03556 

.34889 

2.86624 

46 

15 

.29147 

3.43084 

.31051 

3.22053 

.32975 

3.03260 

.34922 

2.86356 

45 

16 

.29179 

3.42713 

.31083 

3.21722 

.33007 

3.02963 

.34954 

2.86089 

44 

17 

.29210 

3.42343 

.31115 

3.21392 

.33040 

3.02667 

.34987 

2.85822 

43 

18 

.29242 

3.41973 

.31147 

3.21063 

.33072 

3.02372 

.35020 

2.85555 

42 

19 

.29274 

3.41604 

.31178 

3.20734 

.33104 

3.02077 

.35052 

2.85289 

41 

20 

.29305 

3.41236 

.31210 

3.20406 

.33136 

3.01783 

.35085 

2.85023 

40 

21 

.29337 

3.40869 

.31242 

3.20079 

.331C9 

3.01489 

.35118 

2.84758 

39 

22 

.29368 

3.40502 

.31274 

3.19752 

.33201 

3.01196 

.35150 

2.84494 

38 

23 

.29400 

3.40136 

.31306 

3.19426 

.33283 

3.00903 

.35183 

2.84229 

37 

24 

.29432 

3.39771 

.31338 

3.19100 

.33266 

3.C0C11 

.35216 

2.83965 

36 

25 

.29463 

3.39406 

.31370 

3.18775 

.33298 

3.00319 

.35248 

2.83702 

35 

26 

.29495 

3.39042 

.31402 

3.18451 

.33330 

3.00028 

.35281 

2.83439 

34 

27 

.29526 

3.38679 

.31434 

3.18127 

.33363 

2.99738 

.35314 

2.83176 

33 

28 

.29558 

3.38317 

.31466 

3.17804 

.33395 

2.99447 

.35346 

2.82914 

32 

29 

.29590 

3.37955 

.31498 

3.17481 

.33427 

2.99158 

.35379 

2.82653 

31 

30 

.29621 

3.37594 

.31530 

3.17159 

.33460 

2.98868 

.35412 

2.82391 

30 

31 

.29653 

3.37234 

.31562 

3.16838 

.33492 

2.98580 

.35445 

2.82130 

29 

32 

.29685 

3.36375 

.31594 

3.16517 

.33524 

2.98292 

.35477 

2.81870 

28 

33 

.29716 

3.36516 

.31626 

3.16197 

.33557 

2.98004 

.35510 

2.81610 

27 

34 

.29748 

3.36158 

.31658 

3.15877 

.33589 

2.97717 

.35543 

2.81350 

26 

35 

.29780 

3.35800 

.31690 

3.15558 

.33621 

2.97430 

.35576 

2.81091 

25 

36 

.29811 

3.35443 

.31722 

3.15240 

.33654 

2.97144 

.35608 

2.80833 

24 

37 

.29843 

3.35087 

.31754 

3.14922 

.33686 

2.96858 

.35641 

2.80574 

23 

38 

.29875 

3.34732 

.31786 

3.14605 

.33718 

2.96573 

.35674 

2.80316 

22 

39 

.29906 

3.34377 

.31818 

3.14288 

.33751 

2.96288 

.35707 

2.80059 

21 

40 

.29938 

3.34023 

.31850 

3.13972 

.33783 

2.96004 

.35740 

2.79802 

20 

41 

.29970 

3.33670 

.31882 

3.13656 

.33816 

2.95721 

.35772 

2.79545 

19 

42 

.30001 

3.33317 

.31914 

3.13341 

.33848 

2.95437 

.35805 

2.79289 

18 

43 

.30033 

3.32965 

.31946 

3.13027 

.33881 

2.95155 

.35838 

2.79033 

17 

44 

.30065 

3.32614 

.31978 

3.12713 

.33913 

2.94872 

.35871 

2.78778 

16 

45 

.30097 

3.32264 

.32010 

3.12400 

.33945 

2.94591 

.35904 

2.78523 

15 

46 

.30123 

3.31914 

.32042 

3.12087 

.33978 

2.94309 

.35937 

2.78269 

14 

47 

.301G0 

3.31565 

.32074 

3.11775 

.34010 

2.94028 

.35969 

2.78014 

13 

48 

.30192 

3.31216 

.32106 

3.11464 

.34043 

2.93748 

.36002 

2.77761 

12 

49 

.30224 

3.30868 

.32f39 

3.11153 

.34075 

2.93468 

.36035 

2.77507 

11 

50 

.30255 

3.30521 

.32171 

3.10842 

.34108 

2.93189 

.36068 

2.77254 

10 

51 

.30287 

3.30174 

.32203 

3.10532 

.34140 

2.92910 

.36101 

2.77002 

9 

52 

.30319 

3.29829 

.32235 

3.10223 

.34173 

2.92632 

.36134 

2.76750 

8 

53 

.30351 

3.29483 

.32267 

3.09914 

.34205 

2.92354 

.36167 

2.76498 

7 

54 

.30382 

3.29139 

.32299 

3.09606 

.34238 

2.92076 

.36199 

2.76247 

6 

55 

.30414 

3.28795 

.32331 

3.09298 

.34270 

2.91799 

.36232 

2.75996 

5 

56 

.30446 

3.28452 

.32363 

3.08991 

.34303 

2.91523 

.36265 

2.75746 

4 

57 

.30478 

3.28109 

.32396 

3.08685 

.34335 

2.91246 

.36298 

2.75496 

3 

58 

.30509 

3.27767 

.32428 

3.08379 

.34368 

2.90971 

.36331 

2.75246 

2 

59 

.30541 

3.27426 

.32460 

3.08073 

.34400 

2.90696 

.36364 

2.74997 

1 

60 

.30573 

3.27085 

.32492 

3.07768 

.34433 

2.90421 

.36397 

2.74748 

0 

/ 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 



o 

CO 

72° 

71° 

o 

0 



92 




























































natural tangents and cotangents, 



20° 

21° 

22° 

o 

CO 

<M 



Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 


0 

.36397 

2.74748 

.38386 

2.60509 

.40403 

2.47509 

.42447 

2.35585 

60 

1 

.36430 

2.74499 

.38420 

2.60283 

.40436 

2.47302 

.42482 

2.35395 

59 

2 

.36463 

2.74251 

.38453 

2.60057 

.40470 

2.47095 

.42516 

2.35205 

58 

3 

.36496 

2.74004 

.38487 

2.59831 

.40504 

2.46888 

.42551 

2.35015 

57 

4 

.36529 

2.73756 

.38520 

2.59606 

.40538 

2.46682 

.42585 

2.34825 

56 

5 

.36562 

2.73509 

.38553 

2.59381 

.40572 

2.46476 

.42619 

2.34636 

55 

6 

.36595 

2.73263 

.38587 

2.59156 

.40606 

2.46270 

.42654 

2.34447 

54 

7 

.36628 

2.73017 

.38620 

2.58932 

.40640 

2.46065 

.42688 

2.34258 

53 

8 

.36661 

2.72771 

.38654 

2.58708 

.40674 

2.45860 

.42722 

2.34069 

52 

9 

.38694 

2.72526 

.38687 

2.58484 

.40707 

2.45655 

.42757 

2.33881 

51 

10 

.36727 

2.72281 

.38721 

2.58261 

.40741 

2.45451 

.42791 

2.33693 

50 

11 

.36760 

2.72036 

.38754 

2.58038 

.40775 

2.45246 

.42826 

2.33505 

49 

12 

.36793 

2.71792 

.38787 

2.57815 

.40809 

2.45043 

.42800 

2.33317 

48 

13 

.36826 

2.71548 

.38821 

2.57593 

.40843 

2.44839 

.42894 

2.33130 

47 

14 

.36859 

2.71305 

.38854 

2.57371 

.40877 

2.44636 

.42929 

2.32943 

46 

15 

.36892 

2.71062 

.38888 

2.57150 

.40911 

2.44433 

.42963 

2.32756 

45 

16 

.36925 

2.70819 

.38921 

2.56928 

.40945 

2.44230 

.42998 

2.32570 

44 

17 

.33953 

2.70577 

.38955 

2.56707 

.40979 

2.4-1027 

.43032 

2.32383 

43 

18 

.33991 

2.70335 

.33983 

2.56487 

.41013 

2.43825 

.430G7 

2.32197 

42 

19 

.37024 

2.70094 

.39022 

2.56266 

.41047 

2.43623 

.43101 

2.32012 

41 

20 

.37057 

2.69853 

.39055 

2.56046 

.41081 

2.43422 

.43136 

2.31826 

40 

21 

.37090 

2.69612 

.39089 

2.55827 

.41115 

2.43220 

.43170 

2.31641 

39 

22 

.37123 

2.69371 

.39122 

2.556C8 

.41149 

2.43019 

.43205 

2.31456 

38 

23 

.37157 

2.69131 

.39156 

2.55389 

.41183 

2.42319 

.43239 

2.31271 

37 

24 

.37190 

2.68892 

.39190 

2.55170 

.41217 

2.42618 

.43274 

2.31086 

36 

25 

.37223 

2.68653 

.39223 

2.54952 

.41251 

2.42418 

.43308 

2.30902 

35 

26 

.37256 

2.68414 

.39257 

2.54734 

.41235 

2.42218 

.43343 

2.30718 

34 

27 

.37239 

2.68175 

.39290 

2.54516 

.41319 

2.42019 

.43378 

2.30534 

33 

28 

.37322 

2.67937 

.39324 

2.54299 

.41353 

2.41019 

.43412 

2.30351 

32 

29 

.37355 

2.67700 

.39357 

2.54082 

.41387 

2.41G20 

.43447 

2.30167 

31 

30 

.37388 

2.67462 

.39391 

2.53865 

.41421 

2.41421 

.43481 

2.29984 

30 

31 

.37422 

2.67225 

.39425 

2.53648 

.41455 

2.41223 

.43516 

2.29801 

29 

32 

.37455 

2.63939 

.39458 

2.53432 

.41490 

2.41025 

.43550 

2 29619 

28 

33 

.37488 

2.66752 

.39492 

2.53217 

.41524 

2.40827 

.43585 

2.29437 

27 

34 

.37521 

2.66516 

.39528 

2.53001 

.41558 

2.40629 

.43620 

2.29254 

26 

35 

.37554 

2.66281 

.39559 

2.52786 

.41592 

2.40432 

.43654 

2.29073 

25 

36 

.37588 

2.66046 

.39593 

2.52571 

.41626 

2.40235 

.43689 

2.28891 

24 

37 

.37621 

2.65311 

.39626 

2.52357 

.41600 

2.40033 

.43724 

2.28710 

23 

38 

.37654 

2.65576 

.39660 

2.52142 

.41694 

2.39841 

.43758 

2.28528 

22 

39 

.37687 

2.65342 

.39694 

2.51929 

.41728 

2.39645 

.43793 

2.28348 

21 

40 

.37720 

2.65109 

.39727 

2.51715 

.41763 

2.39449 

.43828 

2.28167 

20 

41 

.37754 

2.64875 

.39761 

2.51502 

.41797 

2.39253 

.43862 

2.27987 

19 

42 

.37787 

2.64642 

.39795 

2.51289 

.41831 

2.39058 

.43897 

2.27806 

18 

43 

.37820 

2.64410 

.39829 

2.51076 

.41865 

2.38863 

.43932 

2.27626 

17 

44 

.37853 

2.64177 

.39862 

2 50864 

41899 

2.38668 

.43966 

2.27447 

16 

45 

.37887 

2.63945 

.39896 

2.50652 

.41933 

2.38473 

.44001 

2.27267 

15 

46 

.37920 

2.63714 

.39930 

2.50440 

.41968 

2.38279 

.44036 

2.27088 

14 

47 

.37953 

2.63483 

.39963 

2.50229 

.42002 

2.38084 

.44071 

2.26909 

13 

48 

37986 

2.63252 

.39997 

2.50018 

.42036 

2.37891 

.44105 

2.26730 

12 

49 

.38020 

2.63021 

.40031 

2.49807 

.42070 

2.37697 

.44140 

2.26552 

11 

50 

.38053 

2.62791 

.40065 

2.49597 

.42105 

2.37504 

.44175 

2.26374 

10 

51 

38086 

2.62561 

.40098 

2.49386 

.42139 

2.37311 

.44210 

2.26196 

9 

52 

.38120 

2.62332 

.40132 

2.49177 

.42173 

2.37118 

.44244 

2.26018 

8 

53 

38153 

2.62103 

.40166 

2.48967 

.42207 

2.36925 

.44279 

2.25840 

7 

54 

.38186 

2.61874 

.40200 

2.48758 

.42242 

2.36733 

.44314 

2.25663 

6 

55 

.38220 

2.61646 

.40234 

2.48549 

.42276 

2.36541 

.44349 

2.25486 

5 

56 

.38253 

2.61418 

.40267 

2 48340 

.42310 

2.36349 

.44384 

2.25309 

4 

57 

.38286 

2.61190 

.40301 

2.48132 

.42345 

2.36158 

.44418 

2.25132 

3 

58 

.33320 

2.60963 

.40335 

2.47924 

.42379 

2.35967 

.44453 

2.24956 

2 

59 

38353 

2.60736 

.40369 

2.47716 

.42413 

2.35776 

.44488 

2.24780 

1 

60 

38386 

2.60509 

.40403 

2.47509 

.42447 

2.35585 

.44523 

2.24604 

0 


Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

/ 


69 ° 

68 ° 

67 ° 

66 ° 



93 






































•NATURAL TANGENTS AND COTANGENTS. 


/ 

24 ° 

1 25 ° 

!' 26° 

27 ° 



Tang 

Cotang 

Tang 

Cotang 

1 Tang 

Cotang 

Tang 

Cotang 

0 

0 

.44523 

2.24604 

.46631 

2.14451 

| .48773 

2.05030 

.50953 

1.96261 

60 

1 

.44558 

2.24428 

.46666 

2.14288 

I .48809 

2.04879 

.50989 

1.96120 

59 

2 

.44593 

2.24252 

i .46702 

2.14125 

.48845 

2.04728 

.51026 

1.95979 

58 

3 

.44627 

2.24077 

.46737 

2.13963 

.48881 

2.04577 

.51063 

1.95838 

57 

4 

.44663 

2.23902 

.46772 

2.13801 

.48917 

2.04426 

.51099 

1.95698 

56 

5 

.44697 

2.23727 

1 .46803 

2.13639 

.48953 

2.04276 

.51136 

1.95557 

55 

6 

.44732 

2.23553 

.46343 

2.13477 

.48989 

2.04125 

.51173 

1.95417 

54 

7 

.44767 

2.23378 

.46879 

2.13316 

.49026 

2.03975 

.51209 

1.95277 

53 

8 

.44802 

2.23204 

.46914 

2.13154 

.49062 

2.03825 

.51246 

1.95137 

52 

9 

.44837 

2.23030 

.46950 

2.12993 

.49098 

2.03675 

.51283 

1.94997 

51 

10 

.44872 

2.22857 

.46985 

2.12832 

.49134 

2.03526 

.51319 

1.94858 

50 

11 

.44907 

2.22683 

.47021 

2.12671 

.49170 

2.03376 

.51356 

1.94718 

49 

12 

.44942 

2.22510 

.47056 

2.12511 

.49206 

2.03227 

.51393 

1.94579 

48 

13 

.44977 

2.22337 

.47092 

2.12350 

.49242 

2.03078 

.51430 

1.94440 

47 

14 

.45012 

2.22164 

.47128 

2.12190 

.49278 

2.02929 

.51467 

1.94301 

46 

15 

.45047 

2.21992 

.47163 

2.12030 

.49315 

2.02780 

.51503 

1.94162 

45 

16 

.45082 

2.21819 

.47199 

2.11871 

.49351 

2.02631 

.51540 

1.94023 

44 

17 

.45117 

2.21647 

.47234 

2.11711 

.49387 

2.02483 

.51577 

1.93885 

43 

18 

.45152 

2.21475 

.47270 

2.11552 

.49423 

2.02335 

.51614 

1.93746 

42 

19 

.45187 

2.21304 

.47305 

2.11392 

.49459 

2.02187 

.51651 

1.93608 

41 

20 

.45222 

2.21132 

.47341 

2.11233 

.49495 

2.02039 

.51688 

1.93470 

40 

21 

.45257 

2.20961 

.47377 

2.11075 

.49532 

2.01891 

.51724 

1.93332 

39 

22 

.45292 

2.20790 

.47412 

2.10916 

.49568 

2.01743 

.51761 

1.93195 

38 

23 

.45327 

2.20619 

.47448 

2.10758 

.49604 

2.01596 

.51798 

1.93057 

37 

24 

.45362 

2.20449 

.47483 

2.10600 

.49640 

2.01449 

.51835 

1.92920 

36 

25 

.45397 

2.20278 

.47519 

2.10442 

.49677 

2.01302 

.51872 

1.92782 

35 

26 

.45432 

2.20108 

.47555 

2.10284 

.49713 

2.01155 

.51909 

1.92645 

34 

27 

.45467 

2.19938 

.47590 

2.10126 

.49749 

2.01008 

.51946 

1.92508 

33 

28 

.45502 

2.19769 

.47626 

2.09969 

.49786 

2.00862 

.51983 

1.92371 

32 

29 

.45538 

2.19599 

.47662 

2.C9811 

.49822 

2.00715 

.52020 

1.92235 

31 

30 

.45573 

2.19430 

.47698 

2.09654 

.49858 

2.00569 

.52057 

1.92098 

30 

31 

.45608 

2.19261 

.47733 

2.09498 

.49894 

2.00423 

.52094 

1.91962 

29 

32 

.45643 

2.19092 

.47769 

2.09341 

.49931 

2.00277 

.52131 

1.91826 

28 

33 

.45678 

2.18923 

.47805 

2.09184 

.49967 

2.00131 

.52168 

1.91690 

27 

34 

.45713 

2.18755 

.47840 

2.09028 

.50004 

1.99986 

.52205 

1.91554 

26 

35 

.45748 

2.18587 

.47876 

2.08872 

.50040 

1.99841 

.52242 

1.91418 

25 

36 

.45784 

2.18419 

.47912 

2.08716 

.50076 

1.99695 

.52279 

1.91282 

24 

37 

.45819 

2.18251 

.47948 

2.08560 

.50113 

1.99550 

.52316 

1.91147 

23 

38 

.45854 

2.18084 

.47984 

2.03405 

.50149 

1.99406 

.52353 

1.91012 

22 

39 

.45889 

2.17916 

.48019 

2.08250 

.50185 

1.99261 

.52390 

1.90876 

21 

40 

.45924 

2.17749 

.48055 

2.08094 

.50222 

1.99116 

.52427 

1.90741 

20 

41 

.45960 

2.17582 

.48091 

2.07939 

.50258 

1.98972 

.52464 

1.90607 

19 

42 

.45995 

2.17416 

.43127 

2.07785 

.50295 

1.98828 

.52501 

1.90472 

18 

43 

.46030 

2.17249 

.48163 

2.07630 

.50331 

1.98684 

.52538 

1.90337 

17 

44 

.46065 

2.17083 

.48198 

2.07476 

.50368 

1.98540 

.52575 

1.90203 

16 

45 

.46101 

2.16917 

.48234 

2.07321 

.50404 

1.98396 

.52613 

1.90069 

15 

46 

.46136 

2.16751 

.48270 

2.07167 

.50441 

1.98253 

.52650 

1.89935 

14 

47 

.46171 

2.16585 

.48306 

2.07014 

.50477 

1.98110 

.52687 

1.89801 

13 

48 

.46206 

2.16420 

.48342 

2.06860 

.50514 

1.97966 

.52724 

1.89667 

12 

49 

.46242 

2.16255 

.48378 

2.06706 

.50550 

1.97823 

.52761 

1.89533 

11 

50 

.46277 

2.16090 

.48414 

2.06553 

.50587 

1.97681 

.52798 

1.89400 

10 

51 

.46312 

2.15925 

.48450 

2.06400 

.50623 

1.97538 

.52836 

1.89266 

9 

52 

.46348 

2.15760 

.48486 

2.06247 

.50660 

1.97395 

.52873 

1.89133 

8 

53 

.46383 

2.15596 

.48521 

2.06094 

.50696 

1.97253 

.52910 

1.89000 

7 

54 

.46418 

2.15432 

.48557 

2.05942 

.50733 

1.97111 

.52947 

1.88867 

6 

55 

.46454 

2.15268 

.48593 

2.05790 

.50769 

1.96969 

.52985 

1.88734 

5 

56 

.46489 

2.15104 

.48629 

2.05637 

.50806 

1.96827 

.53022 

1.88602 

4 

57 

.46525 

2.14940 

.48665 

2.05485 

.50843 

1.96685 

.53059 

1.88469 

3 

58 

.46560 

2.14777 

.48701 

2.05333 

.50879 

1.96544 

.53096 

1.88337 

2 

59 

.46595 

2.14614 

.48737 

2.05182 

.50916 

1.96402 

.53134 

1.88205 

1 

60 

.46631 

2.14451 

.48773 

2.05030 

.50953 

1.96261 

.53171 

1.88073 

0 

/ 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

/ 


65 ° 

64 ° 

63 ° 

62 ® 



94 


























































Natural tangents and cotangents, 


/ 

t© 

oo 

0 

29° 

CO 

o 

o 

31° 



Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

| Cotang 

9 

0 

.53171 

1.88073 

.55431 

1.80405 

.57735 

1.73205 

.60086 

1.66428 

60 

1 

.53208 

1.87941 

.55469 

1.80281 

.57774 

1.73089 

.60126 

1.66318 

59 

2 

.53246 

1.87809 

.55507 

1.80158 

.57813 

1.72973 

.60165 

1.66209 

58 

3 

.53283 

1.87677 

.55545 

1.80034 

.57851 

1.72857 

.60205 

1.66099 

57 

4 

.53320 

1.87546 

.55583 

1.79911 

.57890 

1.72741 

.60245 

1.65990 

56 

5 

.53358 

1.87415 

.55621 

1.79788 

.57929 

1.72625 

.60284 

1.65881 

55 

6 

.53395 

1.87283 

.55659 

1.79665 

.57968 

1.72509 

.60324 

1.65772 

54 

7 

.53432 

1.87152 

.55697 

1.79542 

.58007 

1.72393 

.60364 

1.65663 

53 

8 

.53470 

1.87021 

.55736 

1.79419 

.58046 

1.72278 

.60403 

1.65554 

52 

9 

.53507 

1.86891 

.55774 

1.79296 

.58085 

1.72163 

.60443 

1.65445 

51 

10 

.53545 

1.86760 

.55812 

1.79174 

.58124 

1.72047 

.60483 

1.65337 

50 

11 

.53582 

1.86630 

.55850 

1.79051 

.58162 

1.71932 

.60522 

1.65228 

49 

12 

.53620 

1.86499 

.55888 

1.78929 

.58201 

1.71817 

.60562 

1.65120 

48 

13 

.53657 

1.86369 

.55926 

1.78807 

.58240 

1.71702 

.60602 

1.65011 

47 

14 

.53694 

1.86239 

.55964 

1.78685 

.58279 

1.71588 

.60642 

1.64903 

46 

15 

.53732 

1.86109 

.56003 

1.78563 

.58318 

1.71473 

.60681 

1.64795 

45 

16 

.53769 

1.85979 

.56041 

1.78441 

.58357 

1.71358 

.60721 

1.64687 

44 

17 

.53807 

1.85850 

.56079 

1.78319 

.58396 

1.71244 

.60761 

1.64579 

43 

18 

.53844 

1.85720 

.56117 

1.78198 

.58435 

1.71129 

.60801 

1.64471 

42 

19 

.53882 

1.85591 

.56156 

1.78077 

.58474 

1.71015 

.60841 

1.64363 

41 

20 

.53920 

1.85462 

.56194 

1.77955 

.58513 

1.70901 

.60881 

1.64256 

40 

21 

.53957 

1.85333 

.56232 

1.77834 

.58552 

1.70787 

.60921 

1.64148 

39 

22 

.53995 

1.85204 

.56270 

1.77713 

.58591 

1.70673 

.60960 

1.64041 

38 

23 

.54032 

1.85075 

.56309 

1.77592 

.5863*1 

1.70560 

.61000 

1.63934 

37 

24 

.54070 

1.84946 

.56347 

1.77471 

.58670 

1.70446 

.61040 

1.63826 

36 

25 

.54107 

1.84818 

.56385 

1.77351 

.58709 

1.70332 

.61080 

1.63719 

35 

26 

.54145 

1.84689 

.56424 

1.77230 

.58748 

1.70219 

.61120 

1.63612 

34 

27 

.54183 

1.84561 

.56462 

1.77110 

.58787 

1.70106 

.61160 

1.63505 

33 

28 

.54220 

1.84433 

.56501 

1.76990 

.58826 

1.69992 

.61200 

1.63398 

32 

29 

.54258 

1.84305 

.56539 

1.76869 

.58865 

1.69879 

.61240 

1.63292 

31 

30 

.54296 

1.84177 

.56577 

1.76749 

.58905 

1.69766 

.61280 

1.63185 

30 

31 

.54333 

1.84049 

.56616 

1.76629 

.58944 

1.69653 

.61320 

1.63079 

29 

32 

.54371 

1.83922 

.56654 

1.76510 

.58983 

1.69541 

.61360 

1.62972 

28 

33 

.54409 

1.83794 

.56693 

1.76390 

.59022 

1.69428 

.61400 

1.62866 

27 

34 

.54446 

1.83667 

.56731 

1.76271 

.59061 

1.69316 

.61440 

1.62760 

26 

35 

.54484 

1.83540 

.56769 

1.76151 

.59101 

1.69203 

.61480 

1.62654 

25 

36 

.54522 

1.83413 

.56808 

1.76032 

.59140 

1.69091 

.61520 

1.62548 

24 

37 

.54560 

1.83286 

.56846 

1.75913 

.59179 

1.68979 

.61561 

1.62442 

23 

38 

.54597 

1.83159 

.56885 

1.75794 

.59218 

1.68866 

.61601 

1.62336 

22 

39 

.54635 

1.83033 

.56923 

1.75675 

.59258 

1.68754 

.61641 

1.62230 

21 i 

40 

.54673 

1.82906 

.56962 

1.75556 

.59297 

1.68643 

.61681 

1.62125 

20 

41 

.54711 

1.82780 

.57000 

1.75437' 

.59336 

1.68531 

.61721 

1.62019 

19 

42 

.54748 

1.82654 

.57039 

1.75319 

.59376 

1.68419 

.61761 

1.61914 

18 

43 

.54786 

1.82528 

.57078 

1.75200 

.59415 

1.68308 

.61801 

1.61808 

17 

44 

.54824 

1.82402 

.57116 

1.75082 

.59454 

1.68196 

.61842 

1.61703 

16 

45 

.54862 

1.82276 

.57155 

1.74964 

.59494 

1.68085 

.61882 

1.61598 

15 

46 

.54900 

1.82150 

.57193 

1.74846 

.59533 

1.67974 

.61922 

1.61493 

14 

47 

.54938 

1.82025 

.57232 

1.74728 

.59573 

1.67863 

.61962 

1.61388 

13 

48 

.54975 

1.81899 

.57271 

1.74610 

.59612 

1.67752 

.62003 

1.61283 

12 

49 

.55013 

1.81774 

.57309 

1.74492 

.59651 

1.67641 

.62043 

1.61179 

11 

50 

.55051 

1.81649 

.57348 

1.74375 

.59691 

1.67530 

.62083 

1.61074 

10 

51 

.55089 

1.81524 

.57386 

1.74257 

.59730 

1.67419 

.62124 

1.60970 

9 

52 

.55127 

1.81399 

.57425 

1.74140 

.59770 

1.67309 

.62164 

1.60865 

8 

53 

.55165 

1.81274 

.57464 

1.74022 

.59809 

1.67198 

.62204 

1.60761 

7 

54 

.55203 

1.81150 

.57503 

1.73905 

.59849 

1.67088 

.62245 

1.60657 

6 

55 

.55241 

1.81025 

.57541 

1.73788 

.59888 

1.66978 

.62285 

1.60553 

5 

56 

.55279 

1.80901 

.57580 

1.73671 

.59928 

1.66867 

.62325 

1.60449 

4 

57 

.55317 

1.80777 

.57619 

1.73555 

.59967 

1.66757 

.62366 

1.60345 

3 

58 

.55355 

1.80653 

.57657 

1.73438 

.60007 

1.66647 

.62406 

1.60241 

2 

59 

.55393 

1.80529 

.57696 

1.73321 

.60046 

1.66538 

.62446 

1.60137 

1 

60 

.55431 

1.80405 

.57735 

1.73205 

.60086 | 

1.66428 

.62487 

1.60033 

0 

t 

Cotang 

Tang 

Cotang 

Tang 

Cotang | 

Tang 

Cotang 

Tang 

/ 


0 

CD 

60° 

59* 

o 

CO 

•o 



95 















































NATURAL TANGENTS AND COTANGENTS. 



3<2° 

o 

CO 

CO 

CO 

0 

35° 



Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 


0 

.62487 

1.60033 

.64941 

1.53986 

.67451 

1.48256 

.70021 

1.42815 

60 

1 

.62527 

1.59930 

.64982 

1.53888 

.67493 

1.48163 

.70064 

1.42726 

59 

2 

.62568 

1.59826 

.65024 

1.53791 

.67536 

1.48070 

.70107 

1.42638 

58 

3 

.62608 

1.59723 

.65065 

1.53693 

.67578 

1.47977 

.70151 

1.42550 

57 

4 

.62649 

1.59620 

.65106 

1.53595 

.67620 

1.47885 

.70194 

1.42462 

56 

5 

.62689 

1.59517 

.65148 

1.53497 

.67663 

1.47792 

.70238 

1.42374 

55 

6 

.62730 

1.59414 

.65189 

1.53400 

.67705 

1.47699 

.70281 

1.42286 

54 

7 

.62770 

1.59311 

.65231 

1.53302 

.67748 

1.47607 

.70325 

1.42198 

53 

8 

.62811 

1.59208 

.65272 

1.53205 

.67790 

1.47514 

.70368 

1.42110 

52 

9 

.62852 

1.59105 

.65314 

1.53107 

.67832 

1.47422 

.70412 

1.42022 

51 

10 

.62892 

1.59002 

.65355 

1.53010 

.67875 

1.47330 

.70455 

1.41934 

50 

11 

.62933 

1.58900 

.65397 

1.52913 

.67917 

1.47238 

.70499 

1.41847 

49 

12 

.62973 

1.58797 

.65438 

1.52816 

.67960 

1.47146 

.70542 

1.41759 

48 

13 

.63014 

1.58695 

.65480 

1.52719 

.68002 

1.47053 

.70586 

1.41672 

47 

14 

.63055 

1.58593 

.65521 

1.52622 

.68045 

1.46962 

.70629 

1.41584 

46 

15 

.63095 

1.58490 

.65563 

1.52525 

.680S8 

1.46870 

.70673 

1.41497 

45 

16 

.63136 

1.58388 

.65604 

1.52429 

.68130 

1.46778 

.70717 

1.41409 

44 

17 

.63177 

1.58286 

.65646 

1.52332 

.68173 

1.46686 

.70760 

1.41322 

43 

18 

.63217 

1.58184 

.65688 

1.52235 

.68215 

1.46595 

.70804 

1.41235 

42 

19 

.63258 

1.58083 

.65729 

1.52139 

.68258 

1.46503 

.70848 

1.41148 

41 

20 

.63299 

1.57981 

.65771 

1.52043 

.08301 

1.46411 

.70891 

1.41061 

40 

21 

.63340 

1.57879 

.65813 

1.51946 

.68343 

1.46320 

.70935 

1.40974 

39 

22 

.63380 

1.57778 

.65854 

1.51850 

.08386 

1.46229 

.70979 

1.40887 

38 

23 

.63421 

1.57676 

.65896 

1.51754 

.68429 

1.46137 

.71023 

1.40800 

37 

24 

.63462 

1.57575 

.65938 

1.51658 

.68471 

1.46046 

.71066 

1.40714 

36 

25 

.63503 

1.57474 

.65980 

1.51562 

.68514 

1.45955 

.71110 

1.40627 

35 

26 

.63544 

1.57372 

.66021 

1.51466 

.68557 

1.45864 

.71154 

1.40540 

34 

27 

.63584 

1.57271 

.66063 

1.51370 

.68600 

1.45773 

.71198 

1.40454 

33 

28 

.63625 

1.57170 

.66105 

1.51275 

.68642 

1.45682 

.71242 

1.40367 

32 

29 

.63666 

1.57069 

.66147 

1.51179 

.68685 

1 .45592 

.71285 

1.40281 

31 

30 

.63707 

1.56969 

.66189 

1.51084 

.68728 

1.45501 

.71329 

1.40195 

30 

31 

.63748 

1.56868 

.66230 

1.50988 

.68771 

1.45410 

.71373 

1.40109 

29 

32 

.63789 

1.56767 

.66272 

1.50893 

.68814 

1.45320 

.71417 

1.40022 

28 

33 

.63830 

1.56667 

.66314 

1.50797 

.68857 

1.45229 

.71461 

1.39936 

27 

34 

.63871 

1.56566 

.66356 

1.50702 

.68900 

1.45139 

.71505 

1.39850 

26 

35 

.63912 

1.56466 

.66398 

1.50607 

.68942 

1.45049 

.71549 

1.39764 

25 

36 

.63953 

1.56366 

.66440 

1.50512 

.68985 

1.44958 

.71593 

1.39679 

24 

37 

.63994 

1.56265 

.66482 

1.50417 

.69028 

1.44868 

.71637 

1.39593 

23 

38 

.64035 

1.56165 

.66524 

1.50322 

.69071 

1.44778 

.71681 

1.39507 

22 

39 

.64076 

1.56065 

.66566 

1.50228 

.69114 

1.44688 

.71725 

1.39421 

21 

40 

.64117 

1.55966 

.66608 

1.50133 

.69157 

1.44598 

.71769 

1.39336 

20 

41 

.64158 

1.55866 

.66650 

1.50038 

.69200 

1.44508 

.71813 

1.39250 

19 

42 

.64199 

1.55766 

.66692 

1.49944 

.69243 

1.44418 

.71857 

1.39165 

18 

43 

.64240 

1.55666 

.66734 

1.49849 

.69286 

1.44329 

.71901 

1.39079 

17 

44 

.64281 

1.55567 

.66776 

1.49755 

.69329 

1.44239 

.71946 

1.38994 

16 

45 

.64322 

1.55467 

.66818 

1.49661 

.69372 

1.44149 

.71990 

1.38909 

15 

46 

.64363 

1.55368 

.66860 

1.49566 

.69416 

1.44060 

.72034 

1.38824 

14 

47 

.64404 

1.55269 

.66902 

1.49472 

.69459 

1.43970 

.72078 

1.38738 

13 

48 

.64446 

1.55170 

.66944 

1.49378 

.69502 

1.43881 

.72122 

1.38653 

12 

49 

.64487 

1.55071 

.66986 

1.49284 

.69545 

1.43792 

.72167 

1.38568 

11 

50 

.64528 

1.54972 

.67028 

1.49190 

.69588 

1.43703 

.72211 

1.38484 

10 

51 

.64569 

1.54873 

.67071 

1.49097 

.69631 

1.43614 

.72255 

1.38399 

9 

52 

.64610 

1.54774 

.67113 

1.49003 

.69675 

1.43525 

.72299 

1.38314 

8 

53 

.64652 

1.54675 

.67155 

1.48909 

.69718 

1.43436 

.72344 

1.38229 

7 

54 

.64693 

1.54576 

.67197 

1.48816 

.69761 

1.43347 

.72388 

1.38145 

6 

55 

.64734 

1.54478 

.67239 

1.48722 

.69804 

1.43258 

.72432 

1.38060 

5 

56 

.64775 

1.54379 

.67282 

1.48629 

.69847 

1.43169 

.72477 

1.37976 

4 

57 

.64817 

1.54281 

.67324 

1.48536 

.69891 

1.43080 

.72521 

1.37891 

3 

58 

.64858 

1.54183 

.67366 

1.48442 

.69934 

1.42992 

.72565 

1.37807 

2 

59 

.64899 

1.54085 

.67409 

1.48349 

.69977 

1.42903 

.72610 

1.37722 

1 

60 

.64941 

1.53986 

.67451 

1.48256 

.70021 

1.42815 

.72654 

1.37638 

0 

/ 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 



57° 

56° 

55° 

54° 



96 














































Natural tangents and cotangents. 


■* 


/ 

36° 

! 

CO 

<i 

0 

CO 

00 

o 

I 39° 

— 


Tang 

Cotang 

Tang 

Cotang 

Tang 

| Cotang 

Tang 

Cotang 

/ 

0 

.72654 

1.37638 

.75355 

1.32704 

.78129 

1.27994 

.80978 

1.23490 

60 

r 

.72699 

1.37554 

.75401 

1.32624 

.78175 

1.27917 

.81027 

1.23416 

59 

2 

.72743 

1.37470 

.75447 

1.32544 

.78222 

1.27841 

.81075 

1.23343 

58 

3 

.72788 

1.37386 

.75492 

1.32464 

.78269 

1.27764 

.81123 

1.23270 

57 

4 

.72832 

1.37302 

.75538 

1.32384 

.78316 

1.27688 

.81171 

1.23196 

56 

5 

.72877 

1.37218 

.75584 

1.32304 

.78363 

1.27611 

.81220 

1.23123 

55 

6 

.72921 

1.37134 

.75629 

1.32224 

.78410 

1.27535 

.81268 

1.23050 

54 

7 

.72966 

1.37050 

.75675 

1.32144 

.78457 

1.27458 

.81316 

1.22977 

53 

8 

.73010 

1.36967 

.75721 

1.32064 

.78504 

1.27382 

.81364 

1.22904 

52 

9 

.73055 

1.36883 

.75767 

1.31984 

.78551 

1.27306 

.81413 

1.22831 

51 

10 

.73100 

1.36800 

.75812 

1.31904 

.78598 

1.27230 

.81461 

1.22758 

50 

11 

.73144 

1.36716 

.75858 

1.31825 

.78645 

1.27153 

.81510 

1.22685 

49 

12 

.73189 

1.36633 

.75904 

1.31745 

.78692 

1.27077 

.81558 

1.22612 

48 

13 

.73234 

1.36549 

.75950 

1.31666 

.78739 

1.27001 

.81606 

1.22539 

47 

14 

.73278 

1.36466 

.75996 

1.31586 

.78786 

1.26925 

.81655 

1.22467 

46 

15 

.73323 

1.36383 

.76042 

1.31507 

.78834 

1.26849 

.81703 

1.22394 

45 

16 

.73368 

1.36300 

.76088 

1.31427 

.78881 

1.26774 

.81752 

1.22321 

44 

17 

A <3413 

1.36217 

.76134 

1.31348 

.78928 

1.26698 

.81800 

1.22249 

43 

18 

.73457 

1.36134 

.76180 

1.31269 

.78975 

1.26622 

.81849 

1.22176 

42 

19 

.73502 

1.36051 

.76226 

1.31190 

.79022 

1.26546 

.81898 

1.22104 

41 

20 

.73547 

1.35968 

.76272 

1.31110 

.79070 

1.26471 

.81946 

1.22031 

40 

21 

.73592 

1.35885 

.76318 

1.31031 

.79117 

1.26395 

.81995 

1.21959 

39 

22 

.73637 

1.35802 

.76364 

1.30952 

.79164 

1.26319 

.82044 

1.21886 

38 

23 

.73681 

1.35719 

.76410 

1.30873 

.79212 

1.26244 

.82092 

1.21814 

37 

24 

.73726 

1.35637 

.76456 

1.30795 

.79259 

1.26169 

.82141 

1.21742 

36 

25 

•VS ii 1 

1.35554 

.76502 

1.30716 

.79306 

1.26093 

.82190 

1.21670 

35 

26 

.73816 

1.35472 

.76548 

1.30637 

.79354 

1.26018 

.82238 

1.21598 

34 

27 

.73861 

1.35389 

.76594 

1.30558 

.79401 

1.25943 

.82287 

1.21526 

33 

28 

.73906 

1.35307 

.76640 

1.30480 

.79449 

1.25867 

.82336 

1.21454 

32 

29 

.73951 

1.35224 

.76686 

1.30401 

.79496 

1.25792 

.82385 

1.21382 

31 

30 

.73996 

1.35142 

.76733 

1.30323 

.79544 

1.25717 

.82434 

1.21310 

30 

31 

.74041 

1.35060 

.76779 

1.30244 

.79591 

1.25642 

.82483 

1.21238 

29 

32 

.74086 

1.34978 

.76825 

1.30166 

.79639 

1.25567 

.82531 

1.21166 

28 

33 

.74131 

1.34896 

.76871 

1.30087 

.79686 

1.25492 

.82580 

1.21094 

27 

34 

.74176 

1.34814 

.76918 

1.30009 

.79734 

1.25417 

.82629 

1.21023 

26 

35 

.74221 

1.34732 

.76964 

1.29931 

.79781 

1.25343 

.82678 

1.20951 

25 

36 

.74267 

1.34650 

.77010 

1.29853 

.79829 

1.25268 

.82727 

1.20879 

24 

37 

.74312 

1.34568 

.77057 

1.29775 

.79877 

1.25193 

.82776 

1.20808 

23 

38 

.74357 

1.34487 

.77103 

1.29696 

.79924 

1.25118 

.82825 

1.20736 

22 

39 

.74402 

1.34405 

.77149 

1.29618 

.79972 

1.25044 

.82874 

1.20665 

21 

40 

.74447 

1.34323 

.77196 

1.29541 

.80020 

1.24969 

.82923 

1.20593 

20 

41 

.74492 

1.34242 

.77242 

1.29463 

.80067 

1.24895 

.82972 

1.20522 

19 

42 

.74538 

1.34160 

.77289 

1.29385 

.80115 

1.24820 

.83022 

1.20451 

18 

43 

.74583 

1.34079 

.77335 

1.29307 

.80163 

1.24746 

.83071 

1.20379 

17 

44 

.74628 

1.33998 

.77382 

1.29229 

.80211 

1.24672 

.83120 

1,20308 

16 

45 

.74674 

1.33916 

.77428 

1.29152 

.80258 

1.24597 

.83169 

1.20237 

15 

46 

.74719 

1.33835 

.77475 

1.29074 

.80306 

1.24523 

.83218 

1.20166 

14 

47 

.74764 

1.33754 

.77521 

1.28997 

.80354 

1.24449 

.83268 

1.20095 

13 

48 

.74810 

1.33673 

.77568 

1.28919 

.80402 

1.24375 

.83317 

1.20024 

12 

49 

.74855 

1.33592 

.77615 

1.28842 

.80450 

1.24301 

.83366 

1.19953 

11 

50 

.74900 

1.33511 

.77661 

1.28764 

.80498 

1.24227 

.83415 

1.19882 

10 

51 

.74946 

1.33430 

.77708 

1.28687 

.80546 

1.24153 

.83465 

1.19811 

9 

52 

.74991 

1.33349 

.77754 

1.28610 

.80594 

1.24079 

.83514 

1.19740 

8 

53 

.75037 

1.33268 

.77801 

1.28533 

.80642 

1.24005 

.83564 

1.19669 

7 

54 

.75082 

1.33187 

.77848 

1.28456 

.80690 

1.23931 

.83613 

1.19599 

6 

55 

.75128 

1.33107 

.77895 

1.28379 

.80738 

1.23858 

.83662 

1.19528 

5 

56 

.75173 

1.33026 

.77941 

1.28302 

.80786 

1.23784 

.83712 

1.19457 

4 

57 

.75219 

1.32946 

.77988 

1.28225 

.80834 

1.23710 

.83761 

1.19387 

3 

58 

.75264 

1.32865 

.78035 

1.28148 

.80882 

1.23637 

.83811 

1.19316 

2 

59 

.75310 

1.32785 

.78082 

1.28071 

.80930 

1.23563 

.83860 

1.19246 

1 

60 

.75355 

1.32704 

.78129 

1.27994 

.80978 

1.23490 

.83910 

1.19175 

0 

/ 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 



53° 

52° 

51° 

50° 



97 










































.NATURAL TANGENTS AND COTANGENTS. 



O 

o 

41 ° | 

42 ° 

43 ° 


$ 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 


"o 

.83910 

1.19175 

.86929 

1.15037 

.90040 

1.11061 

.93252 

1.07237 

60 

1 

.83960 

1.19105 

.86980 

1.14969 

.90093 

1.10996 

.93306 

1.07174 

59 

2 

.84009 

1.19035 

.87031 

1.14902 

.90146 

1.10931 

.93360 

1.07112 

58 

3 

.84059 

1.18964 

.87082 

1.14834 

.90199 

1.10867 

.93415 

1.07049 

57 

4 

.84108 

1.18894 

.87133 

1.14767 

.90251 

1.10802 

.93469 

1.06987 

56 

5 

.84158 

1.18824 

.87184 

1.14699 

.90304 

1.10737 

.93524 

1.06925 

55 

6 

.84208 

1.18754 

.87236 

1.14632 

.90357 

1.10672 

.93578 

1.06862 

54 

7 

.84258 

1.18684 

.87287 

1.14565 

.90410 

1.10607 

.93633 

1.06800 

53 

8 

.84307 

1.18614 

.87338 

1.14498 

.90463 

1.10543 

.93688 

1.06738 

52 

9 

.84357 

1.18544 

.87389 

1.14430 

.90516 

1.10478 

.93742 

1.06676 

51 

10 

.84407 

1.18474 

.87441 

1.14363 

.90569 

1.10414 

.93797 

1.06613 

50 

11 

.84457 

1.18404 

.87492 

1.14296 

.90621 

1.10349 

.93852 

1.06551 

49 

12 

.84507 

1.18334 

.87543 

1.14229 

.90674 

1.10285 

.93906 

1.06489 

48 

13 

.84556 

1.18264 

.87595 

1.14162 

.90727 

1.10220 

.93961 

1.06427 

47 

14 

.84606 

1.18194 

.87646 

1.14095 

.90781 

1.10156 

.94016 

1.06365 

46 

15 

.84656 

1.18125 

.87698 

1.14028 

.90834 

1.10091 

.94071 

1.06303 

45 

16 

.84706 

1.18055 

.87749 

1.13961 

.90887 

1.10027 

.94125 

1.06241 

44 

17 

.84756 

1.17986 

.87801 

1.13894 

.90940 

1.09963 

.94180 

1.06179 

43 

18 

.84806 

1.17916 

.87852 

1.13828 

.90993 

1.09899 

.94235 

1.06117 

42 

19 

.84856 

1.17846 

.87904 

1.13761 

.91046 

1.09834 

.94290 

1.06056 

41 

20 

.84906 

1.17777 

.87955 

1.13694 

.91099 

1.09770 

.94345 

1.05994 

40 

21 

.84956 

1.17708 

.88007 

1.13627 

.91153 

1.09706 

.94400 

1.05932 

39 

22 

.85006 

1.17638 

.88059 

1.13561 

.91206 

1.09642 

.94455 

1.05870 

38 

23 

.85057 

1.17569 

.88110 

1.13494 

.91259 

1.09578 

.94510 

1.05809 

37 

24 

.85107 

1.17500 

.88162 

1.13428 

.91313 

1.09514 

.94565 

1.05747 

36 

25 

.85157 

1.17430 

.88214 

1.13361 

.91366 

1.09450 

.94620 

1.05685 

35 

26 

.85207 

1.17361 

.88265 

1.13295 

.91419 

1.09386 

.94676 

1.05624 

34 

27 

.85257 

1.17292 

.88317 

1.13228 

.91473 

1.09322 

.94731 

1.05562 

33 

28 

.85308 

1.17223 

.88369 

1.13162 

.91526 

1.09258 

.94786 

1.05501 

32 

29 

.85358 

1.17154 

.88421 

1.13096 

.91580 

1.09195 

.94841 

1.05439 

31 

30 

.85408 

1.17085 

.88473 

1.13029 

.91633 

1.09131 

.94896 

1.05378 

30 

31 

.85458 

1.17016 

.88524 

1.12963 

.91687 

1.09067 

.94952 

1.05317 

29 

32 

.85509 

1.16947 

.88576 

1.12897 

.91740 

1.09003 

.95007 

1.05255 

28 

33 

.85559 

1.16878 

.88628 

1.12831 

.91794 

1.08940 

.95062 

1.05194 

27 

34 

.85609 

1.16809 

.88680 

1.12765 

.91847 

1.08876 

.95118 

1.05133 

26 

35 

.85660 

1.16741 

.88732 

1.12699 

.91901 

1.08813 

.95173 

1.05072 

25 

36 

.85710 

1.16672 

.88784 

1.12633 

.91955 

1.08749 

.95229 

1.05010 

24 

37 

.85761 

1.16603 

.88836 

1.12567 

.92008 

1.08686 

.95284 

1.04949 

23 

38 

.85811 

1.16535 

.88888 

1.12501 

.92062 

1.08622 

.95340 

1.04888 

22 

39 

.85862 

1.16466 

.88940 

1.12435 

.92116 

1.08559 

.95395 

1.04827 

21 

40 

.85912 

1.16398 

.88992 

1.12369 

.92170 

1.08496 

.95451 

1.04766 

20 

41 

.85963 

1.16329 

.89045 

1.12303 

.92224 

1.08432 

.95506 

1.04705 

19 

42 

.86014 

1.16261 

.89097 

1.12238 

.92277 

1.08369 

.95562 

1.04644 

18 

43 

.86064 

1.16192 

.89149 

1.12172 

.92331 

1.08306 

.95618 

1.04583 

17 

44 

.86115 

1.16124 

.89201 

1.12106 

.92385 

1.08243 

.95673 

1.04522 

16 

45 

.86166 

1.16056 

.89253 

1.12041 

.92439 

1.08179 

.95729 

1.04461 

15 

46 

.86216 

1.15987 

.89306 

1.11975 

.92493 

1.08116 

.95785 

1.04401 

14 

47 

.86267 

1.15919 

.89358 

1.11909 

.92547 

1.08053 

.95841 

1.04340 

13 

48 

.86318 

1.15851 

.89410 

1.11844 

.92601 

1 .07990 

.95897 

1.04279 

12 

49 

.86368 

1.15783 

.89463 

1 11778 

.92655 

1.07927 

.95952 

1.04218 

11 

50 

.86419 

1.15715 

.89515 

1.11713 

.92709 

1.07864 

.96008 

1.04158 

10 

51 

.86470 

1.15647 

.89567 

1.11648 

.92763 

1.07801 

.96064 

1.04097 

9 

52 

.86521 

1.15579 

.89620 

1.11582 

.92817 

1.07738 

.96120 

1.04036 

8 

53 

.86572 

1.15511 

.89672 

1.11517 

.92872 

1.07676 

.96176 

1.03976 

7 

54 

.86623 

1.15443 

.89725 

1.11452 

.92926 

1.07613 

.96232 

1.03915 

6 

55 

.86674 

1.15375 

.89777 

1.11387 

.92980 

1.07550 

.96288 

1.03855 

5 

56 

.86725 

1.15308 

.89830 

1.11321 

.93034 

1.07487 

.96344 

1.03794 

4 

57 

.86776 

1.15240 

.89883 

1.11256 

.93088 

1.07425 

.96400 

1.03734 

3 

58 

.86827 

1.15172 

.89935 

1.11191 

.93143 

1.07362 

.96457 

1.03674 

2 

59 

.86878 

1.15104 

.89988 

1.11126 

.93197 

1.07299 

.96513 

1.03613 

1 

60 

.86929 

1.15037 

.90040 

1.11061 

.93252 

1.07237 

.96569 

1.03553 

0 

/ 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

9 


49 ° 

48 ° 

47 ° 

46 ° 



96 









































NATURAL TANGENTS AND COTANGENTS. 


t 

44 ° 

9 

9 

44 ° 

r 

t 

44 ° 


Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

0 

.96569 

1.03553 

60 

20 

.97700 

1.02355 

40 

40 

.98843 

1.01170 

20 

1 

.96625 

1.03493 

59 

21 

.97756 

1.02295 

39 

41 

.98901 

1.01112 

19 

2 

.96681 

1.03433 

58 

22 

.97813 

1.02236 

38 

42 

.98958 

1.01053 

18 

3 

.96738 

1.03372 

57 

23 

.97870 

1.02176 

37 

43 

.99016 

1.00994 

17 

4 

.96794 

1.03312 

56 

24 

.97927 

1.02117 

36 

44 

.99073 

1.00935 

16 

5 

.96850 

1.03252 

55 

25 

.97984 

1.02057 

35 

45 

.99131 

1.00876 

15 

6 

.96907 

1.03192 

54 

26 

.98041 

1.01998 

34 

46 

.99189 

1.00818 

14 

7 

.96963 

1.03132 

53 

27 

.98098 

1.01939 

33 

47 

.99247 

1.00759 

13 

8 

.97020 

1.03072 

52 

23 

.98155 

1.01879 

32 

48 

.99304 

1.00701 

12 

9 

.97076 

1.03012 

51 

29 

.98213 

1.01820 

31 

49 

.99362 

1.00642 

11 

10 

.97133 

1.02952 

50 

30 

.98270 

1.01761 

30 

50 

.99420 

1.00583 

10 

11 

.97189 

1.02892 

49 

31 

.98327 

1.01702 

29 

51 

.99478 

1.00525 

9 

12 

.97246 

1.02832 

48 

32 

.98384 

1.01642 

28 

52 

.99536 

1.00467 

8 

13 

.97302 

1.02772 

47 

33 

.98441 

1.01583 

27 

53 

.99594 

1.00408 

7 

14 

.97359 

1.02713 

46 

34 

.98499 

1.01524 

26 

54 

.99652 

1.00350 

6 

15 

.97416 

1.02653 

45 

35 

.98556 

1.01465 

25 

55 

.99710 

1.00291 

5 

16 

.97472 

1.02593 

44 

36 

.98613 

1.01406 

24 

56 

.99768 

1.00233 

4 

17 

.97529 

1.02533 

43 

37 

.98671 

1.01347 

23 

57 

.99826 

1.00175 

3 

18 

.97586 

1.02474 

42 

38 

.98728 

1.01288 

22 

58 

.99884 

1.00116 

2 

19 

.97643 

1.02414 

41 

39 

.98786 

1.01229 

21 

59 

.99942 

1.00058 

1 

20 

.97700 

1.02355 

40 

40 

.98843 

1.01170 

20 

60 

1.00000 

1.00000 

0 


Cotang 

Tang 

9 

/ 

Cotang 

Tang 

t 

t 

Cotang 

Tang 

f 


45 ° 



45 ° 



45 * 



> > 
) -> > 


99 







































* 




























■*e 


♦ 

























% 







































* 














V 





















. 






' - 



FIELD BOOK 

FOR 

RAILROAD SURVEYING 


BY 

CHARLES LEE CRANDALL, C. E. 

n ’ 

Professor of Bailroad Engineering, Cornell University; 
Member of the American Society of 
Civil Engineers 

AND 

FRED ASA BARNES, C. E. 

Assistant Professor of Bailroad Engineering, Cornell 
University; Associate Member of the American 
Society of Civil Engineers 


THIBD EDITION, ENLABGEB 

FIRST THOUSAND 


NEW YORK 

JOHN WILEY r & SONS 
London: CHAPMAN & HALL, Limited 
1909 


Copyright, 1909 

BY 

CHARLES L. CRANDALL and FRED ASA BARNES 


24 8 7 0 5 


Stanbope lpress 


F. H. GILSON COMPANY 
BOSTON. U.S.A. 


PREFACE. 


This book is an outgrowth of the notes which have been 
prepared from time to time to supplement the text-books 
used in railroad surveying at Cornell. They were enlarged 
to somewhat their present shape some three or four years 
ago and have since been used in mimeographed form as a 
text; the Transition Curve, with one or two additional tables, 
serving as the field book. 

And while an apology is due for adding to the large list of 
books on this subject already available, it is felt that a book 
more adapted to the needs of the student without neglecting 
the requirements of practice would meet a real need. With 
this in view an effort has been made to bring out practical 
field methods and to give representative problems, leaving 
much of the work and most of the special problems for the 
student. Also, as far as practicable, all matter not directly 
connected with railroad surveying has been excluded. This 
has resulted in a book of convenient size for the pocket when 
combined with the Transition Curve, or Part II. 

Long field computations requiring logarithms are unneces¬ 
sary. For office computations, five-place logs are sufficient 
for all railroad field problems. Those by Gauss and by 
Hussey are the most convenient of any known to the authors. 

Cornell University, Ithaca, N. Y., October, 1909 . 


ill 





CONTENTS. 


INTRODUCTION. 

SEC. PAGE 

1. Railroads. 1 

2. Railroad Engineering. 1 

3. Railroad Location. 2 

4. Railroad Surveying. 2 

5. First Principles. 2 

6. General Field Methods. 3 

7. Accuracy. 4 

Chapter I. Simple Curves. 

8. Simple Curves. 7 

9. Degree of Curve. 8 

10. The Deflection Method for laying out Curves. 10 

11. Form of Field Notes. 12 

12. Typical Curve Problem. 13 

13. To lay out a Curve with Inaccessible PI . 16 

14. To lay out a Curve with Inaccessible PC . 17 

15. To change a Curve so as to end in a Parallel Tan¬ 

gent . 19 

16. To draw a Common Tangent to Two Given Circles 20 

17. Sub-Chords. 23 

18. Metric Curves. 23 

19. Other Methods of laying out Curves. 24 

20. Obstacles to Alinement. 28 

21. Alinement by Trial. 29 

Chapter II. Construction. 

22. Leveling. 32 

23. Projecting the Grade Line on the Profile. 33 

24. Volumes. 34 

25. Rules for Cross-sectioning. 36 


v 



























VI 


CONTENTS 


SEC. PAGE 

26. To take Cross-sections... 37 

27. Staking out Work. % 40 

28. To stake out a Bridge Abutment. 41 

29. Referencing a Line. 43 

30. Track-laying and Monumenting. 44 


Chapter III. Reconnoissance. 

31. Survey for Location. 47 

32. The Reconnoissance. 47 

33. Types of Lines. 48 

34. Office Preparation. 49 

35. Field Methods. 50 

36. The Barometer. 52 

37. Barometric Formulas. 54 

38. Barometric Field Work. 57 

39. The Preliminary Survey. 60 

40. The Location. 62 


Chapter IV. Turnouts. 

41. General....... 64 

42. Stub Switch Turnout from Straight Track. 65 

43. Stub Switch Turnout from Curved Track. % . 66 

44. Stub Switch Turnout with Tangent at Frog. 67 

45. Point Swdtch Turnout from Straight Track. 67 

46. Point Switch Turnout on the Inside of Curved 

Track. 69 

47. Point Switch Turnout on the Outside of Curved 

Track. 71 

48. Point Switch Turnouts by Tables. 72 

49. Three Throw Point Switch Turnout. 73 

50. Turnouts wdthout Instruments, Formulas or Tables 73 


























CONTENTS 


Vll 


Chapter V. Special Problems. 

SEC. PAGE 

51. Middle Ordinates for curving Rails. 76 

52. Correction for Curvature and Refraction in Leveling 77 

53. Elevation of Outer Rail on Curves. 77 

54. Vertical Curves. 79 

55. The Stadia Method. 82 

56. Adjustment of Instruments. 83 

See beginning of Part II for its contents, including 

Tables I to VII. Table VIII, as given there, be¬ 
comes Table XVII below. 


Additional Tables. 


Table VIII. Reduction of the Mercurial Barometer to 

Standard for Temperature. 79 

Table IX. Barometric Leveling. 80 

Table X. Barometric Leveling. Temperature Cor¬ 
rection . 83 

Table XI. Point Switch Turnouts from Straight 

Track. 84 

Table XII. Middle Ordinates for curving Rails in 85 

Inches. 85 

Table XIII. Elevation of Outer Rail in Inches. 85 

Table XIV. Corrections to Horizontal Distances for 

Stadia Work. 86 

Table XV. Cubic Yards per 100 Feet, “Level Cut¬ 
tings”. 87 

Table XVI. Formulas. 92 

Table XVII. Natural Sines, Cosines, Tangents and Co¬ 
tangents . (old) 79 


\ 




























■ • 

. 





RAILROAD SURVEYING. 


INTRODUCTION. 

“ Railroad Engineering is the art of making a dollar earn the most 
interest.” — E. H. McHenry in the Northern Pacific Pules. 

1. Railroads. — Railroads are commercial enterprises, 
their special business being the manufacture and sale of trans¬ 
portation. That is, they are built to earn profits for their 
owners, like all other manufacturing plants. 

The life of all business projects may be divided into three 
more or less definite periods: 

1. Investigation and design (called location in the case of 
a railroad), in which the feasibility of the plan and its value 
as an investment are studied and, if the outlook is satis¬ 
factory, plans are made for construction. 

2. Construction. 

3. Operation. 

Capital is invested during the first two periods and earnings 
are expected during the third, — when the net income equals 
the gross earnings less operating expenses and fixed charges. 

2 . Railroad Engineering'. —Railroad Engineering deals 
with all three periods and its general problem is to make the 
net income as large as possible. 

The gross earnings are limited by the amount of transporta¬ 
tion actually sold, i.e., by the amount of traffic. The prin¬ 
cipal factors affecting this, assuming efficient operation, are 
the number and size of the towns served and the character 
of their inhabitants; the extent of the agricultural, mining 
and manufacturing industries along the line; its accessibility 
to the traffic and its position with respect to competing lines. 
It is obvious that these practically limit the earning capacity 
of a line and depend upon its location. 

The operating expenses are also affected by the location, as 
the cost of carrying traffic will vary with the gradients, dis¬ 
tance, curvature and rise and fall over which it must be taken, 

1 



2 


RAILROAD SURVEYING 


[§ 3 . 

and the cost of maintenance will vary with the character of the 
construction and of the ground on which it is constructed. 

The fixed charges consist principally of taxes and interest 
on cost of construction, which again depend upon the location. 
Thus improvements of gradient or alinement for the purpose of 
decreasing operating expenses may increase the fixed charges 
(interest) by a larger amount, or vice versa. 

3. Railroad Location. — The above brief discussion of 
the interdependence of the three periods and of the items affect¬ 
ing the net income indicates not only the complexity of the 
problem confronting the railroad engineer but also the supreme 
importance of the “ location.” The Locating Engineer must 
be versed in construction and operation in order to be able to 
determine the effect of his work on the net income. 

Even if the broader questions of traffic are studied by the 
higher officials and the termini and most important inter¬ 
mediate controlling points are fixed, he must still knowhow 
much it is worth to his line to decrease gradient and to re¬ 
duce distance, curvature and rise and fall, as well as the 
effect of differences in location on the cost of that portion of 
maintenance which is independent, or nearly so, of the traffic. 

4. Railroad Surveying. —It will be impossible, however, 
to discuss even these minor questions within the scope of this 
book, so they will be left to be taken up in the work on Rail¬ 
road Economics, to which subject they properly belong. 

Attention will then be confined to Railroad Surveying, 
which may be defined as the treatment of surveying opera¬ 
tions incident to the location, construction and operation of a 
railroad. 

5. First Principles. — Railroad Surveying has much in 
common with other surveying, but we have seen that it is only 
a small part of the whole field of Railroad Engineering. Only 
a few of the more elemental principles of the latter will there¬ 
fore have to be discussed with the surveying. The most 
important principle will be given here, as it should be learned 
at the outset of one’s railroad training and never be forgotten 
or neglected. 

This first principle recognizes the fact stated in the first 
paragraph and so aptly put in the quotation heading this 
Introduction. This “ art ” must begin with the survey and it 
demands economical work. Economical survey work requires 


GENERAL FIELD METHODS 


3 


§ 8 .] 

thought and careful planning to make every move count to¬ 
wards securing the desired results, concentrating on the 
essential, and slighting the nonessential when economy will 
result thereby. 

Gross blunders and mistakes are best guarded against by 
avoiding haste, by constantly keeping in mind a broad gen¬ 
eral view of the problem and by applying the criterion of 
reasonableness to all results. Thus results plainly impossible 
should be rejected at once and those which appear unreason¬ 
able should be carefully scrutinized before acceptance. 

Uniform general methods are desirable as tending to econ¬ 
omy and freedom from mistakes, but the engineer should be 
ready to adapt his methods to the local conditions whenever 
economy will result thereby. 

The accuracy of the work should be as great as is compatible 
with cost, that is, it should be increased up to the limit at 
which it begins to appreciably increase the cost; beyond this 
limit it should be increased only as actually required in order 
to secure the results needed. This would exclude inaccuracy 
due to poor methods, lack of skill, or carelessness, as these 
do not tend to reduce cost. On the other hand, it is easy to 
waste time and thought on the unimportant details, to the 
neglect of the important ones and especially of the engineering 
considerations upon which the survey depends. 

6. General Field Methods. — Direct horizontal measure¬ 
ments are usually most convenient for a railroad survey, as 
they allow of placing the stakes which mark the stations at 
the usual and regular distance of 100 feet; it being common 
practice in railroad work to express distances in stations, or 
units of 100 feet. On level or uniformly sloping ground, the 
odd stations are sometimes omitted, thus leaving stakes at 
the even stations only, or 200 feet apart. 

On sharp curves, on rough ground and for instrument points, 
intermediate stakes, called plusses, are frequently required. 
A plus of a half station length or of some multiple of 10 feet 
is more convenient for plotting and computation and should 
be taken if suitable. The front tapeman stops at the plus 
with his flagpole and the zero end of the tape, and is alined 
by the transitman. This leaves the rear tapeman, who should 
be held responsible for all tape readings, to note the plus. He 
can remain at the station and hold the end of the tape for the 


4 


RAILROAD SURVEYING 


[§ 7 . 

next station if the line is straight, or he can come up to the 
plus and hold the remainder of the 100 feet on the tape to 
insure the proper distance between stations. 

Each transit point should have a hub and a guard stake. 
The hub is driven nearly flush with the ground and the point 
accurately marked by a tack. Both line and guard stakes are 
marked with the number of the station or station and plus at 
which they are set. The guard stake should be driven about 
one and a half feet to the left of dhe hub, facing it and inclined 
slightly towards it to prevent rapid weathering. Line stakes 
should face backward and be vertical or inclined backward as 
directed by the engineer in charge. The numbers on the stakes 
should all read downward. 

In measuring on rough or sloping ground, the horizontality 
of the tape is estimated by the tapeman. It is well to sight 
over the compass box of the instrument in passing and note 
where its horizontal plane cuts the ground or other convenient 
object. This levels up the tapeman’s horizon and enables 
him to judge more accurately when the tape is horizontal. 
It should be. remembered that a 100-foot steel tape when 
suspended under a 10- to 12-pound pull will project about 
0.1 foot shorter than its true length, and this allowance should 
be made in setting the front stake if the tape is used normally 
flat or suspended in 50-foot lengths. Reliable taping on 
rough or rolling ground can only be secured by occasionally 
watching the tapemen and correcting their methods if found 
defective. On level ground the tape should be flat, with the 
ends on the ground. In fact the tape should always be kept 
as low as possible, i.e., with at least one end on the ground 
unless intervening obstacles prevent. 

If the ground is too steep for 50-foot lengths, the rear tape- 
man should use a hand level for reliable results. He should 
use a plumb bob, while the front tapeman can use his flagpole. 
This pole should be inches wide, with a level for plumbing 
to secure accurate results in taping and pointing; otherwise 
he should take advantage of a tree or house corner at right 
angles to the line to check his estimate of verticality in taping. 

7. Accuracy.—As will be seen later, small errors in aline- 
ment cause inappreciable errors in distance, so that it is use¬ 
less to spend time in alining the regular line stakes closer than 
an inch. The hubs, however, must be accurately alined and 


ACCURACY 


5 


pointed. It is the practice of many engineers to double¬ 
center all hubs, and of others to double-center hubs on long 
curves or long tangents to eliminate errors of collimation. 

Practice will soon teach one the care required to obtain a 
certain degree of precision. The judgment will, however, be 
aided by the following: 

The accuracy desirable in setting over points, plumbing 
poles, etc., i.e., in centering, is such that the errors from these 
causes shall not be greater than the errors of reading the ver¬ 
nier. This is easily determined by remembering that 

sin 1' = .000291, 

which amounts to about 0.03 foot per 100. Thus an error of 
0.0075 foot, or about ts inch, would cause an error of | min¬ 
ute on a 50-foot sight. Hence to run to half minutes with 
50-foot sights requires care in setting the instrument and very 
great care in sighting at the points if the rods cannot be seen 
nearer to the ground than 4 or 5 feet. In fact flagpoles with 
cross levels and well-defined sight lines are essential for 
rough ground in working to minutes unless the sights can be 
made longer than 50 feet. The maximum length of sight in 
pointing should not exceed 1200 feet. With fairly smooth 
country f-inch steel rods or lj-inch steel-pointed wooden rods 
6 to 8 feet long give good results and are much used. 

By remembering that sin l' = 0.00029 and sin 1° = 0.01745, 
or about 0.03 and 1.75 feet per station, respectively, many 



b-x 
Fig. 1. 


field computations can be made without reference to tables, 
or even the use of paper and pencil, which will greatly facili¬ 
tate the passing of obstacles, making corrections, estimating 
deflection angles, etc. 

The formula for correction to distance due to error of aline- 
ment, or to inclination, is derived as follows: In Fig. 1 

b 2 = a 2 + (b - x) 2 

= a 2 + b 2 - 2 bx + x 2 . 



6 


RAILROAD SURVEYING 


[§ 7 . 


Solving, x = -, or 

2 6 — x 

x = — (nearly).(1) 

For a = 0.1 6 the error due to neglecting x in the denominator 
of the second member is only about 1 in 100 000. 

For a 100-foot tape an error of one foot in alinement or in 
level would make an error of of a foot in distance, while 
for a 50-foot tape the error would be twice as great in a tape 
length or four times as great per 100 feet. 




CHAPTER I. 


Simple Curves. 

8. Simple Curves. —Formerly the alinement of a railroad 
as laid out was made up of simple curves and tangents, the 
former being arcs of circles and the latter straight lines tan¬ 
gent to them. The portions of the straight lines between 



the tangent points of adjacent curves are used for the aline¬ 
ment, not the semi-tangents BE and BF of Fig. 2, which are 
called tangent distances and are generally used as auxiliary 
lines in running the curves. At present the practice is 
quite common to change the curvature gradually from the 
straight line to the simple curve by means of a transition curve 
or spiral. This modification will be considered later. 

7 





8 


RAILROAD SURVEYING 


[§ 9 . 


In changing the direction from one tangent to the next, as 
from A B to BD in Fig. 2, the radius of the curve is usually- 
fixed by external conditions; if not, it would have to be 
assumed to make the problem definite, that is, the turn could 
be made with a curve of any radius, as by curve EF with 
radius EO or by E'F' with radius E'O'. 

For curve EF in Fig. 2, E and F are the tangent points, and, 
assuming the curve to be run from left to right or from E to 
F , the point E is called the point of curve, PC, and the point 
F the point of tangent, PT; the intersection of the tangents, 
B, is called the point of intersection, PI] the deflection angle 
from one tangent to the next is called the intersection or 
central angle, J; the distance from the PI to the tangent 
point (BE or BF) is called the tangent distance, or simply the 
tangent, T; the perpendicular distance from the PI to the 
curve, BG, is called the external distance, E; the chord EF 
joining the tangent points is called the long chord, C, and the 
distance of its center from the curve, NG, is called the middle 
ordinate, M. 

Denoting the radius of the circle by R we have by trigo¬ 
nometry: 

Tangent distance, T = R tan i 4 .... (2) 

External distance, E = R exsec h A . . , . (3) 

Middle ordinate, M = R vers h 4 . . . . (4) 

Long chord, C = 2 R sin i A . . . . (5) 


9. Degree of Curve. — Since the above functions of a 
curve depend directly upon the radius, it seems natural to 
designate a curve by its radius, and this is the English practice. 
In America, however, a curve is designated by its degree for 
greater convenience in laying out by the method of deflections, 
except for very sharp street railroad curves, where the English 
system is frequently followed. 

The degree of curve, D, is the central angle subtended by a 
chord of 100 feet. This would give, Fig. 3, 


which makes the radius vary inversely as D for small angles. 
For a 1° curve, R = 5729.65; for a 10°, R = 573.69; for a 



Eq. 6.] 


DEGREE OF CURVE 


1 


9 


20°, R — 287.94, etc; while the corresponding arcs, or lengths 
of track between 100-foot stations, are, in feet, 100.001 
100.127, 100.510, etc. 

It is customary and convenient to tabulate the tangents and 
externals of a 1° curve as found by (2) and (3) and to find 
those for other curves by dividing by D. These quotients 
require corrections which increase with D and with A. On 
account of this difficulty and because of sharp curves being so 



Fig. 3. 


much longer than their nominal or recorded lengths, it has 
been proposed to make the degree of curve the central angle 
subtended by an arc of 100 feet and to use a table of chord 
corrections in the field, shortening the chord as D increases 
so as to give 100-foot arcs. This method has not come into 
extended use. 

Another method suggested by A. M. Wellington has come 
into more general use. It consists in using shorter chords for 
the sharper curves, 50-foot for curves from 8° to 16°, 25-foot 
from 16° to 32°, and 10-foot for sharper curves, — so that 
the chords will closely approximate the arcs. The radius of 
the 1° curve is changed from 5729.65 to 5730, while that for a 
D° curve is 5730 /D. This method has been followed in the 
Transition Curve of Part II. It is accurate enough for all 
field-work except possibly running track centers, for which 
corrections are given on page 39 of Part II. Sharp curves 
require stakes closer together than 100 feet to properly define 
them, so that the short chords seldom require useless 
labor. 





10 


RAILROAD SURVEYING 


[§ 10 . 


Accepting this modified definition, 

Length of curve in stations, L 

Radius in feet, R 

The radius of a 1° curve can be easily 
ciation with the number of feet (5280) in a mile; the first 
and last digits are the same, as also the sum of the other 
two. 

Tangents and externals for a 1° curve are given in Table VII 
of Part II with A as argument; those for a D° curve are readily 
found by dividing by D. The radii are given in Table V; 
,if D is integral, as will usually be the case, they are as readily 
found for the flatter curves by simple mental division. 

10. The Deflection Method for laying out Curves. — 
The methods of laying out curves are simply those of drawing 
arcs of circles on the ground. When the radius is small, say 
less than 100 feet, the arc may be swung in from its center, 
using the tape. This is sometimes done in street railroad 
work, but ordinarily the deflection method is used on account 
of the impracticability of using the radius. This method 
depends upon the theorem that for a circle the angle between 
a tangent and a chord or between two chords is measured by 
one-half the intercepted arc and that angles at the center 
are measured by the intercepted arcs. Thus with the transit 
at any point of the curve the deflection for 100 feet (measured 
in whole, halves, quarters or tenths of a tape length according 
to the sharpness of the curve) will be D/2 and proportionately 
for other distances. 

Thus, in Fig. 4, suppose that the tangent has been located 
from the left to the PI and that A has been measured. As¬ 
sume a degree of curve, D, that will meet the conditions im¬ 
posed. Then find the radius from Table V or by dividing 
5730 by D, and the tangent distance, T, by (2), or more con¬ 
veniently by dividing the tangent for a 1° curve, T v from Table 
VII by D. Subtract T from the stationing for PI for the 
PC and measure T forward from the PI for the PT. Then 
take the transit back to the PC, assumed as sta. 162 + 25, 
and deflect from the tangent as follows! 


A 

D 

5730 

D 


( 7 ) 

( 8 ) 


remembered by asso- 



Eq. 8.] 


THE DEFLECTION METHOD 


11 


Sta. 163, or A, .75 of £ D. 

Sta. 164, B, .75 of i D + i D. 

Sta. 165, C, .75 of 1 D + D. 

Sta. 166, D, .75 of * D + f D. 

Sta. 167, E, .75 of £ D + 2D. 

Sta. 167 + 30 PT, .75 of i D + 2 D + 0.3 of i D = \ 
as the deflection for the PT is always one-half the central 
angle. This checks all of the deflections, and these notes 
should be written up before running the curve. 



Fig. 4. 


A set of deflections made up in this way may be used with 
the instrument at any point on the curve. Each deflection 
or vernier reading should be associated with the station to 
which it belongs, and it can be used for that station so long as 
the instrument is set up anywhere on the curve. This will 
require the vernier for the backsight to be set at the reading 
opposite the station on which the backsight is taken. The 







RAILROAD SURVEYING 


12 


[§ 11 . 


reading opposite the station where the instrument is set up 
will give the tangent to the circle at that point. 

Obstructions often require the instrument to be moved up 
one or more times in running a curve, while if the central angle 
is large,, accuracy will require a new set-up as soon as the arc 
described by the front end of the tape revolving around the 
last station as a center does not give a large-angled intersec¬ 
tion with the line of sight from the instrument. These new 
instrument points will have no effect on the curve notes or 
vernier readings made up once for all as indicated above, 
unless plus stations are used, in which case the extra vernier 
readings are simply interpolated. 

Many engineers make up a new set of vernier readings for 
each intermediate instrument point on the curve, starting 
from zero on the new tangent. While this system has some 
advantages, it is believed that the method outlined above will 
save labor and reduce the danger of mistakes. 

11. Form of Field Notes. —The form of transit notes 
shown in Fig. 5 is convenient. The O’s in the first column 
show the hubs or transit points on the line. They indicate 
where the line can be accurately picked up for further instru¬ 
ment work, and to quite an extent the character of the work, 
and they should always be given. 

The computations for the 6° curve are given in full in the 
column of Curve Data and no further explanation is required 
except to state that the 167 + 51.6 opposite the PT is the 
stationing around the tangents and is found by adding the tan¬ 
gent, T, to the PI. The track is to be laid around the curve, 
so the stationing 167 + 30 by‘the curve is the one which will 
govern in running the next tangent. 

T u the tangent for a 1° curve for A =36° 18', is taken from 
Table VII as already indicated. 

The bearings noted are read from the needle for each tangent, 
and they should always be given as a rough check on the work 
and for comparison with those of property and other required 
lines. The bearings shown in brackets are computed from the 
A’s of the curves, starting from the first or reference tangent. 
Table V, giving deflection angles in minutes per foot of 
chord, may be conveniently used when the sub-chord deflec¬ 
tion cannot readily be computed mentally. The date, party 
and usually the weather conditions, should be given at the 


Eq. 8.] 


TYPICAL CURVE PROBLEM 


13 


beginning of the day’s work. If the work is not a con¬ 
tinuation of that immediately preceding in the record it should 
be*described. Under this latter head should be given the title 
of the problem if it is problem work. 

12. Typical Curve Problem. —In the typical or standard 
curve problem one tangent has been located and station stakes 
set and the other tangent has been picked out on the ground. 

Set up at the PI, measure A, and note the bearing of the 
front tangent; while the tapemen are chaining for the PT look 
up T , in Table VII, divide by D and give the resulting 
tangent length to the rear tapeman for locating and pointing 
the PT hub; subtract T from PI for the PC; divide A by D 
for length of curve, L, which add to the stationing of the PC 
and give the result to the stakeman, for marking the guard at 
the PT ; check the backsight and give the PC stationing to 
the tapemen so that they can put in the PC hub and guard; 
complete the curve notes, checking the PT through the 
vernier readings as indicated in §11; set up on the PC and 
backsight on the tangent with vernier at zero. By taking up 
the work in this order the transitman can keep his party busy 
and have the notes ready for running the curve as soon as the 
PC is set. 

If the PT is visible from the PC it is well to turn one-half 
A and note the check upon the P T, as this checks the equality 
of the tangents and the value of A. In running the curve, 
the last sub-chord should check the computed value, and the 
line of sight should check the tack on the PT hub. The 
discrepancy, if within allowable limits, is usually left in the 
curve, the front tangent starting off at the P T from a back 
sight on the PI. The same direction should of course be 
obtained by backsighting to the last hub on the curve and 
turning to the vernier reading (j A) for the PT. 

As already stated, as many intermediate instrument stations 
as necessary may be used on the curve without disturbing the 
notes except to enter the O’s and also the extra vernier 
readings in case full stations cannot be utilized. The guards 
at the ends of curves should be marked to show the alinement, 
as PC 3° R, PT, etc. 

If the PI is not given, two stakes may be pointed on the 
rear tangent not over 100 feet apart, and preferably less than 
20, so as to include the PI between them. The transit may 


14 


RAILROAD SURVEYING 


TRANSIT 


Sta. 

Point. 

Curve 

or 

Bear. 

Vern. 

Curv 

j Data 

1 



2-00 



+60.0 c. 

PC 

10°R 




170 












9 G 












8 






+ 80.0 a 

PT 

S 6-20 W 

18-09 

= 167 

+h.6 

7 


[6-28] 

17-15 







aJ 

36-18 

6 



14-15 

II 

cJf§ 

1^ 

= 6.05 





R = 

955 

165 



11-15 

r,= 

1878.4 





T = 

313.1 

4 0 



8-15 

PI= 

164+38.1 





PC= 

161+25.0 

3 



5-15 

PT— 

167+30.0 







2 



2-15 



+25.0 a 

PC 

6°R 

0 



1 











t 

160 © 


S 30-10 E 






[29-50] 











Figure 




































TRANSIT BOOK 


15 


BOOK 



5 






























































16 


RAILROAD SURVEYING 


[§ 13 . 


then be set over the nearer of the two points fixing thfe front 
tangent and the PI found at the intersection of the line of 
collimation for the front tangent with the tape stretched 
between the two stakes on the rear tangent. 

In general, to solve a field problem, a sketch should be 
made, as nearly to scale as may be, putting in the known data 
and then proceeding step by step to determine the required 
data upon the supposition that it is being done by plotting. 
After this is done, follow the plotting steps by trigonometric 
relations or formulas which will furnish a basis for the required 
numerical solution. It is much easier to follow relations 
graphically, and it is usually possible to secure numerical 
results for any problem in field geometry which can be plotted. 
With this habit once acquired, one is independent of field 
books except for the tables, and much better equipped for 
rapidly and accurately seeing the relations and solving the 
problems arising in railroad location or in track work. On 
this account only a few of the special problems which come up 
in connection with simple curves and obstacles in alinement 
will be given, and these to bring out methods rather than on 
account of any difficulty in their solution. 

Long numerical solutions in the field should be avoided so 
far as possible. The delays occasioned are expensive, demor¬ 
alizing to the party and injurious to its reputation. Fre¬ 
quently much of the solution can be done graphically with 
the instrument and party with a saving of time and reputa¬ 
tion; at other times data can be gathered for an office solution 
and other work taken up for the balance of the day. Study 
and experience soon enable one to approximate the best 
obtainable results in fitting a line to the ground by simple 
methods and simple geometric relations, leaving the difficult 
special problem for yards, city streets, or rugged topography. 

13. To lay out a Curve with Inaccessible PI. — From 
a suitable point on the first tangent run a traverse over con¬ 
venient ground to reach a point on the second tangent. These 
points on the tangents should be near the desired tangent 
points, or at least not much nearer the PI than the tangent 
points. Assume the first tangent as a meridian or axis of X 
and compute the coordinates of the points on the traverse 
with reference to this, using the starting point as the origin. 
Divide the ordinate of the station on the second tangent by 


Eq. 9.] 


INACCESSIBLE PC. 


17 


sin A for the distance of the point from the PI and multi¬ 
ply this ordinate by cot A for the projection of this distance 
on the axis. Subtract this projection from the x coordinate 
of the point for the distance of the point on the first tangent 
from the PI. Compare these distances from the PI with 
the tangent of the required curve for the distances to move 
from these points on the tangents to the PC and PT. 
After setting these the remainder of the problem will be the 
same as given in §12. 


Example. — Given at sta. 184, deflection 4° 18' left; sta. 
186 + 50, 2° 10' right; sta. 189, 6° 25' left; sta. 192, 1° 18' left, 
coinciding with the second tangent. Required the data for 
locatingi a 1° curve. 

Solution. 


Station 

Deflection 

Dist. 

X 

y 

184 

4° 

18' 

L 

250 

249.30 

18.74 

186 +50 

2 

8 

L 

250 

249.83 

9.31 

189 

8 

33 

L 

300 

296.67 

44.60 

192 

9 

51 

L = A 


— 


Coordinates of sta. 192, 


795.80 

72.65 


yf 72 65 

Distance of sta. 192 from PI = -+—- = = 424.7. 

sin A .17107 


Projected distance, y' cot A = 72.65 X 5.75941 = 418.42. 
Sta. 184 from PI, 795.80 - 418.42 - 377.38. 

PI = 187 4- 77.4 

T 1 (Table VII) = 493.7 = 4 + 93.7 


PC 

L 9.85 
PT 


= 182 + 83.7 
9+85 
= 192 + 68.7 


The PT will be 493.7 — 424.7 = 69.0 beyond the old sta. 192 . 
14. To lay out a Curve with Inaccessible PC- — 
Extend the tangent to a regular station near the obstruction 
as at A, Fig. 6, and pass by an equilateral triangle with sides 
one or more full stations long, or by a traverse or other 
method suitable for the conditions, and determine the PI 
and A. Find T and PC, and by comparison with CD decide 
upon the station number for G ; subtract the PC from G in 
stations and multiply by D for «. 







18 


RAILROAD SURVEYING 


[§ 14 . 


Set the PT and F while the instrument is at PI and set 
up at F, turn 90° from PI, set G and place a backsight hub H 
on FG produced some 200 feet, if FG is short. Set up at G , 
backsight on H with vernier at 90° — £ a and continue the 
curve from the notes made up as usual. 


D 



A sketch with the special data and computations should 
be shown on the right-hand page. 

A simpler method would be to turn 90- % A from the back¬ 
sight when at PI and measure the external, E, locating the 
central point of the curve. 

The vernier for the backsight from this point would be 
90° — i A; the regular curve notes would answer with an 
interpolated reading for this central point, but the first por¬ 
tion of the curve would have to be backed in and would not be 
checked. 

Example. —Given: — sta. 185, 60° R for two stations; 120° L 
for two stations to sta. 187; sta. 187, 60° R; sta. 189 + 70 PI- 
A - 18° 20' R. 



Eq. 10.] 


PARALLEL TANGENT 


19 


Find the data for a 2° curve and the coordinates for a starting 
point on the curve free of the obstruction. 

qoa 6 

Table VII, Ti = 924.6, Ex = 74.12, T= -—— = 462.3; 

2 

74 12 

E = —— = 37.06; R = 2865. 

4 

18 33 

PC = 185 + 07.7 ;L-— = 9 + 16.7; 

PT = 194 + 24.4. 

For sta. 187 on the curve; a= 1.923 X 2 = 3° 50£'. 

PC to F = 2865 X sin 3° 50£' = 2865 X .06700 = 192.0 
DF = 462.3 - 192.0 = 270.3 
FG = 2865 (1 - cos 3° 50*') = 2865 X .00225 = 
6.45; giving the co°rdinates for sta. 187 of 
the curve. 

15. To change a Curve so as to end in a Parallel 
Tangent. — 

(a) by sliding the PC on the first tangent. If the PC slides 
along the first tangent the PT will slide the same distance 
parallel with the first tangent, so that in the triangle ABD, 
Fig. 7, for a perpendicular distance p between tangents the 
distance AD moved along the first tangent will be found from 

AD = JL. . (10) 

sin A 

This distance AD through which the PC moves is to be 
measured backward if the new tangent is inside the old, as in 
Fig. 7, and forward if the new tangent is outside the old. 

Shifting the position of the curve will change the transit 
notes to correspond with the stationing of the new tangent 
points. 


Example. — Sta. 184 + 50, PC, 4° R; sta. 190, PT. Re¬ 
quired the new tangent point and the vernier readings for running 
the curve if the new tangent is to come 10 feet inside the old. 

By (10), 


AD 


10 

sin 22° 


10 

37461 


26.7. 


New alinement: 184 + 23.3, PC, 4° R; deflections,— 185, 
1° 32'; 186, 3° 32'; 187, 5° 32'; 188, 7° 32'; 189, 9° 32'; 189 + 73.3, 
PT, 11° 00'. 

(6) by changing the degree of the curve. 







20 RAILROAD SURVEYING [§ 16 . 

If the PC remains fixed, the new tangent T' would be less 
than the old by AD, Fig. 7, giving 


T' = T ± . (11) 

sin A 



Fig. 7. 


If the PT is to come opposite the old, 

T' = T T p cot A, (12) 

where the upper sign is for the new tangent outside the old 
and the lower sign for the new tangent inside the old. Having 
T' y 

D' = . (13) 


16 . To draw a Common Tangent to Two Given 
Circles.—This problem may arise in changing the location of 
part of a line or in joining up a new location to old track. 





Eq. 13.] 


COMMON TANGENT 


21 


The curves are connected in the field by stopping the first, at 
such a point that its tangent when extended will cut the 
other. The stationing for this intersection point is found and 
the deflection angle measured with one or more of the chords 
radiating from it. This supplies the data shown in Fig. 8. 



Thus A B is tangent to the first curve at A and intersects 
the second at B with a for the intersection angle. 

In the right triangle ABO, Fig. 8, 


tan O x 
OB 

In the triangle BOO', 


AB 

9 

R 

AB 
sin O x 


angle B 2 = 90 + a — (90 — 0,) 
= a + O v 


(а) 

( б ) 


1 

j 


OB and O'B, or R', are known. 

Substituting for the oblique triangle with two sides and the 
included angle known, 

tan H0 2 - O') = R '~ ° B tan (90 - i B 2 ). (c) 

R' + OB 

O 2 = i (0 2 + O') + (0 2 - O'), 

O' = i(0 2 + O') -i(0 2 -O'), 


00 '_ 

sin 0 2 


(d) 










22 


RAILROAD SURVEYING 


[§ 16 . 


With O' as a center and R' — R as a radius describe a circle. 
A tangent to this circle through 0 will be parallel to the 
required tangent. 

To find OE and the angle data: 

In the right triangle OEO' 

. n R' - R 

sm 0 3 = -> 

00 ' 

OE = 00' cos 0 3 , 
t 8 = 90 -f- 0 3 — (Oj “t - 0 2 ). 

The distance to move the PT on the first curve, 

GA = (3/D. 


For the PC at F on the second curve, add OE to the sta¬ 
tioning for the PT at G. 

There are three other cases which are left for the student; 
one with the centers on the opposite sides of the required 
tangent and the other two with the flat curve coming first. 

Unless a traverse is required in passing from A to B on 
account of obstructions, this problem can be more easily 
solved in the field by locating A by §21 so near the required 
PT that the angle between AB and GF will be so small that 
it may be converted into distance on the curve by dividing 
by D. 

If a traverse is required (or available) to give the relative 
positions of the two curves, the problem is readily solved by 
computing coordinates, referred to a convenient axis, joining 
the curve centers and computing the necessary distances and 
angles as above. 

Example. — The tangent at sta. 162, the PT of a 6° L curve 
intersects a 2° curve at sta. 168 with a = 4° 18' R. Required the 
PT of the 6° and the PC of the 2° for a common tangent. As this 
is an office problem the computation by logs will be given in full. 

(a), AB = 600. 2.77815 (6), AR= 600. 2.77815 

R 6 = 955. 2.98000 O l = 32° 08' sin 9.72582 


Ox = 32° 08'tan. 9.79815 OB = 1128.1_ 3.05233 


a, 4° 18' 
Ox, 32 08 

B 2 , 36 26 


(c), R 2 — OB, 1736.9. 3.23977 

R 2 + OB, 3993.1 (AC) _ 6.39869 

90 - £ B 2 , 71° 47'. . .tan_ 0.48266 

£(0 2 “ O'), 52°53'. .tan_ 0.12112 

0 2 , 124° 40'; O', 18° 54'. 









Eq. 14 .] METRIC CURVES 23 

(d), Ri, 2865 . 3.45712 R 2 - R 6 , 1910... 3.28103 


Bi, 36° 26'sin.. 9.77370 00', 2068.7 3.31570 

O 2 , 124 40 sin (AC)0.08488 Os, 67° 24' sin ... 9.96533 

00', 2068.7.3.31570 


90 

+ O 3 , 

157° 24' 

Os, 

67° 24' cos ... 9.58467 

Ox 

+ 0 2 , 

156 48 

OO 1 

’, 2068.7. 3.31570 

8, 


0 36 

OE 

, 795.0. 2.90037 


0° 

36' 



GA 


« " 0 + 10 - 

New 

FT = 162 - (0 + 10) = 


161 + 90. 

Tangent Distance GF — OE =7 + 95; PC 2° L = 169 + 85. 

17. Sub-Chords. — Referring to §9, it may be noted that 
if curves are computed and run on the basis of 100-foot chords 
a sub-chord length c' for a given deflection angle d would be 
found from (6), 

c' = 2 R sin d, (14) 

with R = 50/sin £ D. 

Or, if the sub-chord c is given, the corresponding deflection 
angle d would be found from 

sin d = . (15) 

2 R 

These values will not differ appreciably from the nominal 
ones for curves flatter than 8°. 

On the basis of shortening the chords as the degree of curve 
increases the true sub-chord would be 

c' = 2 R sin d , (16) 

with R = 5730 /D. 

This will differ so little from the nominal length that no 
correction will be necessary except in cases of extreme ac¬ 
curacy in running track centers. 

18. Metric Curves. —In working under the metric system 
it is customary to use a 20-meter tape and to take the central 
angle subtended by a tape length (for flat curves) as the 
degree of curve. This makes the radius of a metric curve to 
that of the foot curve of the same degree as 65.62 (the unit 
chord) is to 100 (the unit chord), or approximately as 2 to 3, 










24 


RAILROAD SURVEYING 


[§ 19 . 


while there are but one-fifth as many units in the chord, 20 in 
place of 100, and hence in the tangent or any of the linear 
functions of the curve. 

The tables for the English or foot system are thus available 
for the metric by simply dividing all linear functions by 5 
D instead of by D. 

The stations are numbered as if a 10-meter tape were used, 
leaving out the odd-numbered stations except when inter¬ 
mediates are required. 

This gives one-fifth as many units in the curve functions 
and one-tenth as many in the station numbers as in the foot 
system, which is confusing at first, but the confusion soon 
disappears with use. 

In following up the Wellington system for degree of curve 
it would be somewhat on the safe side to change to 10-meter 
chords for from 8° to 16° curves, 5-meter chords from 16° to 
32°, and 2-meter chords for curves sharper than 32°. 

19. Other Methods for layiiig out Curves. — Most of 
these are for use without a transit and are not capable of 
great accuracy except for short curves on smooth ground. 

(a) by offsets from the tangents. This is a method by 
coordinates, using one of the tangents as an axis and the 
tangent point as an origin. In Fig. 9, A is the PC and B, D , E 
and F are regular stations on the curve. 

Let AC = n, a fractional part of a station length. Then 
AB = 1 — n, and the angle 

E'AB = HI ~ n)D, 

AB ' = AB cos Hi - n)D, > 

BB' = AB sin \ (1 — n)D. ) ' •' 

A tangent at B would make an angle of (1 — n)D with the 
original tangent, and the chord BD would make an angle of 
D/2 additional. 

Hence the angle between the chord BD and the tangent 
CE' is (| — n) D, and 

B'D' = BD cos (f — n)D, or 

B'D' = 100 cos (f - n)D, 1 

DD" = 100 sin (| — n)D.\ ( 18 ) 

Similarly, 

DE" = 100 cos (f - n)D, | 

EE" = 100 sin (f — n)D. j 


Eq. 20.J 


TANGENT OFFSETS 


25 


After passing the middle of the curve the offsets and dis¬ 
tances can be found by working from the other tangent if 
more convenient. 

On flat curves the offsets only need be computed and the 
chords can be used with them in locating the stations. Thus 


F" 



Fig. 9. 


the tapemen may often be able to put in a flat curve accurately 
enough for grading the roadbed while the transitman is 
moving forward. 

Another method of computing coordinates is often used. 
It has the advantage of giving the coordinates for each sta¬ 
tion directly and the disadvantage of using a large factor, R, 
as compared with summing the coordinates in the previous 
case and obtaining the increments, after the first, by simply 
pointing off two places in the natural functions. The equa¬ 
tions are AB' = R sin (1 - n) D, 

BB' = R vers (1 — n) D, 

AD' = R sin (2 — n) D, 

DD' = R vers (2 — n ) D. 


( 20 ) 





26 RAILROAD SURVEYING [§ 19 . 

If the tangent point is at a full station, n = 0 and dis¬ 
appears from the formulas. 

Offsets from tangents may be found approximately from 
the following: 

Let n = the distance in stations from the tangent point 
to the point where the offset is desired. Then £ nD will be 
the angle between the tangent and the chord, and 

offset = 100 n sin £ nD. 

Since small angles are proportional to their sines, 
sin nD = .01745 nD° (approx.). 

Substituting, offset from tangent, 

£ = £ n 2 D, (21) 

where n is in stations and the corresponding deflection 
angle is so small that for its sine the angle in degrees into the 
sine of 1° can be used. 



Fig. 10. 


By writing a similar equation for a D' degree curve, and 
subtracting, distance between curves, 

t' - £n 2 (Z> - D'). (22) 

This equation is useful for correcting trial curves or for 
determining the effect of changing D. 

(b) by offsets from chords produced. — If in Fig. 10 the 
perpendicular CF be dropped from C to the tangent BF at 
the next full station B, the tangent offset, CF, or £, will sub¬ 
tend an angle CBF equal D/2. 







CHORD OFFSETS 


27 


Eq. 23.] 

Tangent offset, t — 100 sin \ D, ) 

or t = | D° (approx.), j 

If CF be extended to meet the next 100-foot chord A B 
produced, at E, the chord offset CE , or o, will be twice the 
tangent offset. 

Chord offset, o = 200 sin \ D, 
or o = \ D° (approx.). 

In starting to lay out the curve from the tangent point, 
measure each way to a full station and lay off the tangent 
offsets by (17) or (21); prolong the chord through these two 
station points 100 feet forward and lay off the chord offset to 
locate the next station 100 feet from the front station through 
which the chord was drawn; extend the second chord and 
repeat. 

This method is especially valuable in enabling the front 
tapeman to approximate the line on curves nearly as well as 
on tangents, thus saving time and checking the transitman. 
It is also valuable on construction in replacing the stakes 
approximately for checking the grading, etc., since it is 
usually possible to recover at least one of the chords and the 
curve may be extended either way from it. 

(c) By deflections with two transits. The transits are 
set up at two different points on the curve, for example the 
PC and PT, and each set at the proper vernier reading for 
a required station; the intersection of the two lines of colli- 
mation will locate the station. The central angle of the curve 
between the two instruments should be great enough to give 
a fairly large intersection angle for accurate location. This 
method is sometimes used in locating points in a pond or other 
place where linear measurements are inconvenient. 

These special methods are given to suggest expedients for 
convenience in field work. Others will suggest themselves 
to the thoughtful student as the result of experience. 

The following approximate formulas in this connection 
should be memorized as greatly facilitating field-work. 
t = | n 2 D, 
t' = In 2 (D - D'), 
m = n 2 D, 
o = i D, 



( 25 ) 



RAILROAD SURVEYING 


28 


[§ 20 . 


where t = tangent offset; t' = offset between two curves; 
m — middle ordinate; and o = chord offset. 

20. Obstacles to Alinement. — For transit work the 
deflection method is generally preferable to the offset method 
for passing obstacles as being quicker and more accurate. 

The equilateral triangle, when available, involves no com¬ 
putations; the isosceles triangle involves computation for 
distance only, and this may be avoided by using deflection 
angles which will give simple ratios. 

Thus any one of the following combinations of deflection 
angle and distance will give a projection forward of 100 feet 
and proportionally for other distances. 


Deflection. 

Distance. 

Deflection. 

Distance. 

84° 

M’ 

500 

56° 

15' 

90 

82 

49 

400 

51 

19 

80 

80 

24 

300 

44 

25 

70 

75 

31 £ 

200 

33 

33* 

60 

70 

32 

150 

24 

37 

55 

60 

00 

100 

00 

00 

1 

50 


The traverse with coordinates (example, §13) is perfectly 
general, but it requires computation. For small deflection 
angles from the main direction or axis of X the cosines may 
be taken as unity and the sines proportional to the angles, or 
since the ordinates or offsets are not needed for any inter¬ 
mediate point and the last ordinate is zero, the angles in 
minutes may be used instead of the sines. This gives the 
following: 

Rule. — Call deflections to the right positive, and those 
to the left negative. Multiply the length of each course by 
the algebraic sum of the deflections up to that course and 
add the products. The main line is reached when the algebraic 
sum of these products is zero, and its direction is given when 
the algebraic sum of the deflections is zero. 


Example. — 


Station. 

Deflection. 

Algebraic 


Product. 


R L 

Sum 


+ 

_ . 

106 

0° 50' 

+ 50' 

50 X 950 

47 500 


115 + 50 

1°10' 

-20 

20 X 250 


5 000 

118 

0 30 

-50 

50 X 650 


32 500 

124 + 50 




47 500 

37 500 












Eq. 25.] 


ALINEMENT BY TRIAL 


29 


At sta. 124 + 50 the algebraic sum of the products is + 10000 
and the direction of the line from sta. 118 to sta. 124 + 50 
is 0° 50' L. 

If it is desired to reach the line in 5 stations, it would 

require a deflection of — = o° 20' to the left of the 

500 

main line, or of 0° 50'— 0° 20' = 0° 30' right from the last 
line or backsight, and at sta. 129 +50 a deflection of 0° 20' 
to the right, to continue the original line. Any other con¬ 
venient distance could have been assumed and the correspond¬ 
ing deflections obtained. 

On curves, if care, is taken not to place the transit too close 
to a small obstacle, it can usually be passed by using chords 
which will give clear lines of sight for plus stations close 
enough together to carry the taping along the curve and then 
interpolate for the regular stations, or if this is not feasible, by 
setting a hub at the end of a long chord (computed by (5)), 
and then running the curve backward to the obstacle. 

For larger obstacles one of the general methods already 
described can be applied to a long chord and the curve then 
run backward to the obstacle. 

21. Alinement by Trial. — While the typical method of 
location is to fit the tangents to the ground and then connect 
them by suitable curves, as indicated in §12, the method 
becomes inconvenient when the intersection angles are large, 
as the points of intersection will come quite a distance from 
the located line, and the extended tangents may be on rough 
ground or obstructed, especially if in a timbered country. 

Again, when the curves are long as compared with the 
tangents it is often better to fit them to the ground first 
rather than the tangents. For these reasons and because of 
the saving in time which can be effected, the method of running 
the curve forward from the PC without having the front 
tangent definitely located is much used. With this method 
special care is required to prevent mistakes, as no very pre¬ 
cise checks on the curves are obtainable. The discrepancy 
between the computed and the magnetic bearing would show 
any serious mistake in the instrument work. 

If the curve does not fit the ground it can be moved laterally 
by sliding the PC along the first tangent or by changing the 
degree of the curve as required. The proper point at which 



30 


RAILROAD SURVEYING 


[§ 21 . 

to stop the curve can be determined by pacing off the tangent 
offset at the rear end of the tape and sighting over the stake 
at the front end and noting how the direction agrees with that 
of the required tangent. If it must be determined closer 
than to the nearest station, a plus station can be set tem¬ 
porarily and the corresponding tangent offset (= | n 2 D, or the 
chord X sine of deflection angle) laid off and the line tested 
before bringing up the instrument. 

It is believed that this method is admissible up to 1000 feet 
for long curves, as any discrepancy found in the alinement 
can be readily taken care of by slightly shifting the positions 
of curves or tangents during construction. 


PROBLEMS. 

1. A line is measured along an even slope and found to be 670 
feet long. One end is 50 feet higher than the other. 

Find the true horizontal distance by ( a ) an exact method; 
(6) the approximate method. 

2. A trial line measured through brush is found to be 890 feet 
long and to require an offset of 26 feet to pass through the required 
point. 

Find the true distance by (a) an exact method; (6) the approxi¬ 
mate method. 

3. Given M = 50 ft. and A = 40° 25'. 

Required E, T, R and C, 

4. Given C = 300 ft. and A = 10° 45'. 

Required E, T, D and M. 

6. It is desired to put in the regular stations only on a 14° curve. 
How much must the nominal 100-ft. chords be shortened to 
accomplish this, (a) if 50-ft. chords are to be used later; (6) if 
100-ft. arcs are desired between stations? 

6. A street railway curve of 90-ft. radius is swung in from its 
center. 

Required the degree of the curve, (a) on the 100-ft. chord basis; 
(6) on the 100-ft. arc basis; (c) on the basis of using 10-ft. chords. 

7. A 20° curve is to be run in at first with 100-ft. chords. 
Required the deflection angles for putting the points on a curve 

to be run later with 25-ft. chords. 

8. A curve is run with 5° deflections for 50-ft. chords. 
Required the exact radius and the degree of the curve. 

9. Find the coordinates, using the tangent as the axis of X, for 
locating 4 regular stations on a 5° curve with PC at sta. 180 + 20. 
Compare the ordinates with those found by the approximate 
method. 


Eq. 25.] 


PROBLEMS 


31 


10. Find the coordinates from the long chord for laying out the 
regular stations of a 7° curve with PC at sta. 180 + 60 and PT at 
sta. 185 + 10. 

11. Two tangents with bearings N. 60° 40' E. and S. 80° 30' E. 
intersect at sta. 180 +4,0 metric with A = 39° 05' R. Com¬ 
puted bearing of first tangent [60° 29']. 

Required complete transit notes for a 6° metric curve. 

12. Given A = 36° 15' L; PI = sta. 260+ 6.0, metric/ 
E = 10™. Bearings of tangents N. 70° 20' W. and S. 73° 20'* 
Computed bearing of the first [70° 16']. 

Required the complete transit notes. 

13. Given A = 26° 35' R; PI = sta. 184+ 10. 

Required the PC and D of a curve to pass through a point 
10 feet to the right of sta. 182 on the tangent. 

14. Given the following alinement: — sta. 176, PC 3° R; sta. 
182, PT ; sta. 185 + 50, PC 5° R; sta. 189, PT. 

Required the PCC’s for substituting a 2° curve for the tangent 
between the 3° and 4° curves. 

15. A 1° curve ends at sta. 181. On prolonging the tangent to 
sta. 190 it is found that 1° 30' should be added to the curve to 
bring the tangent on the required ground. 

Find the offset from the old tangent to sta. 182 + 50 and the 
distance and direction from the old sta. 190 to the new sta. 190 
so as to set up on the new sta. 190 and backsight on the new 
182 + 50 and run the new tangent without having to go back 
with the instrument. 

16. A 4° curve is run from sta. 210 to sta. 216. On setting up 
at sta. 216 it is found that a 3° curve with the same PC would 
better fit the ground. 

Find the distance and direction to lay off from the instrument 
to hub sta. 216 on the 3° curve and the vernier reading for sta. 
217 after setting up on the new station and backsighting upon the 
PC. 


CHAPTER /II. 


Construction. 

22. Leveling. —As soon as the center line is located levels 
are run over it to determine the ground elevations. These 
should be taken at all stations and also at all breaks in grade 
in order to fully develop the ground surface. High-water 
marks and other drainage conditions should be carefully 
noted by the levelman, as also the classification or character 
of the soil, as these must be considered in fixing the height 
of the grade line upon the profile. Sea level datum should 
be used if readily obtainable. 

Bench marks should be established at points about one- 
half mile apart for use on location if running a preliminary 
or for use on construction if running a location. They should 
usually be near grade points for convenience during con¬ 
struction but far enough away to avoid disturbance. Roots 
of trees, good masonry and large boulders make excellent 
BM’s for railroad work if properly marked and described; 
when not available, poorer ones have to be used. 

With backsights and frontsights on turning points equal, 
or the sums between bench marks equal, and with sights 
limited to about 400 feet, there should be no difficulty in 
checking levels to less than 0.1 foot per mile on fairly rough 
ground, and there is no advantage worth spending money 
for in checking much closer than this on ordinary railroad 
location. 

The readings on turning points only are placed in the BS 
and FS columns. The difference of the sums of these 
columns should check the last turning point or HI. 

Thus, 

24.86 - 11.37 = 13.49, 

13.49 + 793.38 = 806.87, the last HI. 


32 


Eq. 25.] PROJECTING THE GRADE LINE 
The following is a convenient form for level notes. 


33 


LEVEL BOOK. 


Sta. 

BS. 

HI. 

FS. 

Rod. 

BM. 

11.59 

804.97 



O 

272 

12.61 

817.41 

0.17 



1.3 

273 




1.7 

274 




2.2 

+ 60 




4.1 

275 




5.7 

276 




5.4 

+ 40 




9.0 

+ 50 




12.3 

+ 65 




8.4 

4- 80 




10.0 

o 

+ 90 

0.66 

806.87 

11.20 



1.9 





24.86 


11.37 



Elev. 

Remarks. 

793.38 

Large stump, 

804.80 

816.1 

815.7 

815.2 

813.3 

811.7 
812.0 

808.4 

10 R. of 278. 

805.1 

Stream 3' X 4' 

809.0 

807.4 

806.21 

805.0 

Rock. 


23. Projecting the Grade Line on the Profile.— The 

fundamental condition governing the fixing of the grade line 
is that the cuts shall balance the fills, as this gives minimum 
yardage to be moved and therefore cheapest construction for 
uniform material unless leads are thereby increased. An 
apparent balancing on the profile does not mean an actual 
balancing for the following reasons: — 

1. The width of roadbed in cuts is made greater than on 
fills in order to give room for side ditches and sometimes 
the side slopes are different. 

2. Earth shrinks on an average about 10 per cent from 
cut to fill, and rock swells some 60 to 75 per cent. 

3. Steep transverse slopes increase the yardage for a given 
center height. 

Using tables of level cuttings and making the allowances 
noted, even the novice can “ balance cuts and fills ” in time, 
but considerable experience is required to expeditiously lay 
on a grade line “ by eye ” which will satisfy with sufficient 
accuracy even the fundamental condition given above. 






























34 


RAILROAD SURVEYING 


[§ 24 . 


In fact it is usually unnecessary to accurately satisfy this 
condition, as one or more of the following factors enter into 
the question and good judgment and experience are required 
to strike a proper balance between them. 

Excess excavation, or excess fill, may be required when 
the height of the grade line is modified by having to pass 
under or over an existing railroad or important highway, or 
when a location requiring light grading only would cost 
excessively for right of way. 

Excess excavation is often required at summits in order 
to reduce the gradient to the required limit. 

Embankment must be in excess over low, flat country, sub¬ 
ject to overflow, where the grade line should be kept at least 
two feet above high water. First cost is often reduced by 
raising the grade line to reduce the cuts and using temporary 
trestles in place of the expensive portions of the fills. 

When cuts are in solid rock or hardpan it is often more 
economical to raise the grade line and borrow cheaper material 
for the excess fills. 

Deep cuts should be avoided in regions of heavy snow and 
whenever they would seriously increase the danger from 
landslides. While excess material may generally be used 
to advantage in widening out banks, it is best, as a rule, to 
keep the grade line fairly high, as fills tend to give cheaper 
maintenance on account of better drainage. A piece of fine 
thread is very useful in studying the projection of a grade 
line. 

The grade line of this paragraph is usually called the sub¬ 
grade; it defines the surface of the finished roadbed after 
grading. The ballast, ties and rails are placed on this road¬ 
bed, making the grade of the track a constant distance higher 
than the subgrade. 

24. Volumes. — After fixing the grade line on the profile, 
the grade elevations or “ grades ” are computed for each 
station and, together with the ground elevations, entered 
in the cross-section book. The center cuts or fills are then 
made up as shown in the notes. It is desirable to have also 
a list of M’s in the back of the book. 

Tables of level cuttings could then be used to determine 
volumes if the ground were exactly level transversely. But 
as this rarely happens and sinca it is necessary to set slope 


Eq. 25.] 


VOLUMES 


35 


stakes at the edges of the cuts and fills to guide the con¬ 
tractor in grading, cross-sections are taken to determine the 
exact shape of each section. The cross-section notes are 
then available for accurately computing the volumes for 
which the contractor must be paid. 

The simplest section is the two-level section in which the 
transverse slope is uniform, as in Fig. 11, making it necessary 



to determine the heights at the slope stakes, E and F, only, 
i.e., at but two points. Since the center cut CD (or fill) is 
also known, however, this section is usually treated as a 
three-level section, just as if a break occurred at the center 
as in the true three-level section. 

If the transverse slope breaks over or under the edges of 
the roadbed it is necessary to take readings at these points 



also, giving the five-level section shown in Fig. 12, i.e., one 
in which five points are taken. 

When the transverse slope is still more broken it is necessary 
to take readings at each break, giving an irregular section. 

The volume between adjacent sections is often computed 
by simply multiplying the length between them by the aver¬ 
age of the two end areas. This does not, however, give true 
volume, excepting in those rare cases when the earthwork 
solid is a prism or wedge. In the usual case the earthwork 
solid is a prismoid, which may be defined as a solid bounded 
by bases in parallel planes and whose sides are generated by 













36 


RAILROAD SURVEYING 


[§ 25 . 


lines which move by proportional parts along the homologous 
sides of the figures determining the parallel ends. The 
sides may also be considered as generated by lines constantly 
parallel with the planes of the bases moving along the lines 
connecting homologous points of the ends, as directrices. 
Thus the sides are either planes or hyperbolic paraboloids, 
and the volume of such a solid may be computed by the fol¬ 
lowing : 

V = - (A + A' + 4 A m ), (26) 

6 

where A and A' are the end areas and A m the middle area. 

From an inspection of this formula it is easily seen that 
the average end area method gives correct results only when 
A m is an average of the two end areas. Since it would be a 
tedious operation to find A m by the interpolation of all linear 
dimensions, it is usual to divide the more complex prismoids 
into triangular ones for the purpose of computation, especially 
in case the true volume is to be determined. The volume 
by averaging end areas is then' found first, preferably by the 
use of tables (see Crandall’s Tables for the Computation of 
Railway and Other Earthwork), and then a prismoidal 
correction is determined, also from tables. This correction 
depends upon the change in height and change in base between 
the two triangular ends. 

25. Rules for Cross-sectioning. — From the definition 
of the prismoid it is evident that straight lines between any 
two homologous points of the parallel bases are elements 
of the surface, and this is the criterion which governs the field¬ 
work, and necessitates the following rules for taking cross- 
sections. 

Take cross-sections at all regular stations and at plusses 
where necessary, so that straight lines connecting homolo¬ 
gous points of adjacent sections shall lie wholly in the ground 
surface or equalize that surface. When the center line is on 
a curve, the lines connecting homologous paints must be 
estimated at the same curvature in judging of their coincidence 
with the ground surface. 

Take sufficient points in each cross-section so that straight 
lines joining them will lie in the surface or will equalize the 
surface. 


Eq. 26.] 


TO TAKE CROSS-SECTIONS 


37 


The subdivision into triangular prismoids requires the 
same number of sides or points in adjacent cross-sections, so 
that, when a ridge or hollow runs out, its vanishing point in 
the more regular section must be taken and the section made 
up with the break for the solid on one side and without it 
for the solid on the other side of the section. The height of 
this vanishing point can be computed, but it is better to meas¬ 
ure it in the field. An occasional extra break can often be 
treated best by itself, as adding or subtracting a computed 
amount to or from the regular quantity which would be 
obtained without it. 

Cross-sections should also be taken wherever a grade point 
occurs on either edge of the roadbed (usually limited to the 
edges in cutting), and one is generally placed where a grade 
point occurs on the center line. 

On light work and where the transverse slope is small, 
a section is sometimes taken where the center line comes 
to grade and the other two sections are omitted. 

The plusses are preferably made 50 feet, multiples of 10 
feet, or whole feet (except grade points), for convenience in 
computing grades in the field and quantities in the office. 
Breaks on a cross-section should be put at whole feet from 
the center when possible without a sacrifice of accuracy. 

When the method by averaging end areas is used for 
computing the volumes of the prismoids, additional sections 
should be taken whenever the change in transverse slope or 
in area between adjacent sections is great, i.e., when the 
change in height and width is great, giving a large prismoidal 
correction. 

26* To take Cross-sections. —In taking cross-sections 
some engineers run a new line of levels and work from the 
HI’ s. This is hardly worth while, especially if the BM’s 
have been used on the preliminary survey and rechecked on 
the location. It is desirable, however, to guard against 
local errors by checking between consecutive stations when 
convenient. 

The usual process is to use the center cut or fill as a basis 
and from that determine cuts and fills at such other points 
as are necessary. 

Thus (see form of Cross-section Book) the center cut at 
station 186 is 13.7. The rod is held on center and reads, 


38 


RAILROAD SURVEYING 


[§ 26 . 


CROSS-SECTION 


Sta. 

Elev. 

Grade. 

Cut. 

Fill. 

Rod. 

+50. 

853.8 

869.6 


15.8 


+ 16. 

866.3 

869.5 


3.2 


+ 08. 

869.3 

869.5 


0.2 





CO 



187. 

870.3 

869.5 

° 0.8 



+90. 

872.2 

869.5 

+ 

- 2.7 



+50. 

879.6 

869.3 

10.3 


20.3 

186. 

882.9 

869.2 

13.7 


20.4 


say 6.7. The “ grade rod ” or reading of the rod if it could 
be held at grade would then be 20.4 (see rod column). That 
is, the line of sight is 20.4 above grade. 

Then a reading is taken 10 feet out from center on the left. 
This is found to be 8.3, and subtracting this from the grade 
rod we have 12.1 for the cut at that point, which is noted as 
shown. 

In this section it is next necessary to determine the position 
of the slope stake or where the side slope of the roadbed 
intersects the ground surface, as at E and F, Figs. 11 and 12. 
This point is found by trial. 

In Figs. 11 and 1,2, EB and FA are the slopes whose 
inclination is expressed as the ratio of horizontal to vertical 


projection. 


Thus if— = 
FH 


H 

1 ’ 


the slope is 1^:1, or 1£ hori¬ 


zontal to 1 vertical. 

Thus for a point to be on this slope, the distance from the 
edge of the roadbed must be equal to the cut or fill at the 






























Eq. 26.] 


CROSS-SECTION BOOK 


39 


BOOK. 


Left. 

Center. 

Right. 


- 17.4 

- 16.6 

-15.8 

- 14.0 

- 12.6 

34.1 

8 

8 

26.9 

- 9.7 

- 6.9 

- 3.2 

00 


22.5 

8 

id 


- 7.7 

- 4.5 

- 0.2 

00 +1.1 

+ 1.4 

19.5 

8 

2 10 

12.1 

- 1.0 

- 0.3 00 

+ 0.8 

. + 2.4 

+ 3.3 

9.5 

8 5 

10 

14.9 


00 

4- 2.7 

+ 4.3 

+ 5.8 


10 

10 

18.7 

+ 6.3 

+ 8.1 

+ 10.3 

+ 11.9 

+ 15.8 

19.4 

10 

10 

33.7 

+ 8.8 

+ 12.1 

+ 13.7 

+ 15.0 

+ 19.1 

23.2 

10 

10 

38.6 


point multiplied by the slope ratio. The distances are 
always measured from the center line, which requires the 
half width of roadbed to be added in order to get the total 
distance. 

Thus suppose the rod reads 11.1. at 20 feet from center. 
The cut at that point is then 20.4 — 11.1 =9.3; X 
9.3 = 14.0; 14.0 + 10 = 24.0, which does not check with the 
measured distance, 20 feet. It shows, however, that the 
measured distance should be increased, so other points are 
tried until the correct one is found Thus the rod reading 
is 11.6 at 23.2 feet from center, i.e., the cut is 20.4 — 
11.6 = 8.8; 1* X 8.8 = 13.2; 13.2 + 10 = 23.2, which 

checks the distance out. 

The slope stake is then marked C 8.8 and driven at 23.2 
left of center. These stakes are usually faced towards and 
sloped away from the center line, so that they will not be 
confused with other stakes. 

In like manner the rod is held over the edge of the roadbed 





























40 


RAILROAD SURVEYING 


[§ 27 . 


on the right (10 feet out) and a reading of 5.4 found, giving 
20.4 — 5.4 = 15 for the cut. which is noted as shown. The 
slope stake is then found as before. 

Rapid cross-sectioning requires good judgment in the selec¬ 
tion of points and ability to calculate quickly and mentally. 

Apply the rule for distances out to each of the slope stakes 
given in the notes. The method of checking from one 
station to the next is indicated by the grade rod at 186 + 50. 
The rod reading at center is 10, giving 10 + 10.3 = 20.3 
for grade rod. Comparing the difference of grades with the 
difference of grade rods from same set-up for the adjacent 
sections, they are found to be the same, thus checking the 
work. Sometimes it will, of course, be necessary to carry 
levels over a turning point in order to apply this check. 

The same principles as given above for cuts also apply to 
fills, the only difference being that the instrument may be 
under the grade line instead of over it. 

A grade point occurs on center between 186 + 90 where 
the left side runs out of cut and 187 + 16 where the cut 
ends on the right. The fills do not begin, of course, at 
exactly these same points, but it is usual to consider them 
to do so, as it is not necessary to have the volumes of fills as 
accurately as those of the cuts. 

Grade points along the center line are determined by mov¬ 
ing the rod along the line until the reading equals the grade 
rod at the preceding station plus or minus the correction due 
to the gradient up to the point where the reading is taken, 
depending on whether the gradient is minus or plus. They 
may also be computed approximately from the notes by 
finding the rate of change of the surface. Thus in the notes 
given there is a grade point on center line between 187 and 

0 8 

187 + 08 whose distance from 187 is---x 8 = 6 4 feet 

0.8 + 0.2 

The same process serves to determine grade points at the 
edges of the roadbed except that the rod must be moved 
along a line parallel with the center line and distant from 
it by one-half the width of the roadbed. 

27. Staking out Work. — The ordinary masonry struc¬ 
tures like culverts, bridge abutments, trestle piers, etc., usu¬ 
ally consist of two parts, the foundation and the neat work. 
For convenience, or to increase the bearing area, the founda- 



Eq. 26.] TO STAKE OUT A BRIDGE ABUTMENT 41 

tion is usually larger and the excavation pit is dug the full 
size or more for the foundation. When the foundation is built 
the face lines for the neat work are drawn and the masons or 
carpenters work to them for the portions of the structure in 
sight. If the face walls are battered, allowance must be 
made at the bases so that the tops will be in their proper 
positions. If the structure does not extend up to the road¬ 
bed, the exact distance below the latter is not required in 
laying out the work unless it is limited by the roadbed side 
slopes. In this case the relation between distance out from 
center and depth below grade is determined as for cross- 
sectioning. 

For bridge masonry, trestle piers, etc., the office plans will 
show the height with reference to tops of ties or subgrade 
as the case may be, and this height must be worked to accu¬ 
rately from foundation up or proper allowances made in 
locating the neat work lines on the ground. 

Masons use a string in laying stone to a face line, stretch¬ 
ing it first between the face line stakes given by the engineer 
and afterwards between corner stones set with a batter 
board at the proper slope, 

When it is necessary to extend a curve across a bridge its 
axis should be a chord which will leave the same lateral 
clearance at the center as at the ends and the masonry 
staked with reference to this the same as if the track were 
straight. The bridge may of course require widening on 
account of the curve. 

Great care is required in laying out structures to avoid 
mistakes and inaccuracies. If plans are not furnished, a 
sketch plan should be made in the notebook showing all 
stakes and dimensions or other data used. If plans are 
furnished, the sketch should also be made showing the field 
data. These field data should be reviewed at night as a 
further precaution against mistakes. 

28. To stake out a Bridge Abutment. — The general 
plan and section are shown in Fig. 13. The subgrade eleva¬ 
tion at the left abutment is 196.22, and the bottom of the 
neat work of the abutment is 176.22. The face batter under 
the coping is one inch to a foot. 

Set up on the center line, A B, of the roadbed and point C 
for the base of the neat work face and D for the foundation. 


42 


RAILROAD SURVEYING 


[§ 28 . 


Set up at C, turn 90° and set E and F for the corners of the 
wing walls; extend this face line say 20 feet each way and 
put in solid reference hubs to preserve the face line. Set 
up at E and lay off 30° for the face of the wing wall. If 
staking from a general plan, the end of the wing G can be 
found by making its distance from A B = 1.5 times depth 




below grade, plus half width of roadbed. It is customary to 
make the height of the end of the wing 3 or 4 feet and allow 
the fill to slope slightly around the end. If staking from a 
plan, the length would be given and could be laid off directly. 
Reference hubs can be placed on EG produced at such dis¬ 
tances as not to be disturbed by the excavation. 

Similarly the wing FH can be staked and referenced. 
Having these face lines the foundation lines can be laid off 
from them without an instrument, or they can be laid off 
independently from D. 

The excavation should be carried below frost if on soil, 
while other considerations such as danger of undercutting 
or poor material may require it to be carried much deeper. 




























Eq. 26.] 


REFERENCING A LINE 


43 


Levels are run from the nearest BM and the HI, say 
179.84, found for the last set-up. Then 179.84 — 176.22 = 
3.62, the rod reading for top of foundation masonry. Drive 
substantial stakes, one near the center and one near each 
end, until the rod reads 3.62 when held on the top. These 
stakes should be near enough so that the masons’ level board 
will reach the masonry. 

The face batter will be carried up the quoins, JE and KF , 
and its value should be computed so as to give the 1 inch 
per foot on 1C. 

For coursed masonry the height should be checked from 
time to time with the drawings to insure the bridge seat’s 
coming at the correct height. 

The distance Cl in inches = height of battered face in 
feet; the clear span under the coping will be greater than that 
at the base by twice Cl. 

If the span is long, or there are several spans between the 
abutments, it is necessary that the tape agree with the bridge 
company’s standard for the structure to fit the masonry. 

29. Referencing a Line. — During construction practic¬ 
ally all the stakes and hubs on the center line are dug out or 
covered. This requires that the line be referenced for ease 
of recovery in running track centers or even for giving line 
and grade as may be necessary from time to time during the 
progress of the work. 

If the P/’s are outside the slope stakes and accessible they 
are excellent points to preserve by substantial hubs. When 
these are not suitably situated, the PC’s and P/’s may be 
referenced to hubs or other objects outside the slope stakes. 
This may be done by measuring a distance each way at right 
angles to the line, or by measuring two distances on the 
same side. If there is a deep cut or high fill to be made at 
the point, the remeasurements will be inconvenient and the 
exact recovery difficult. 

On this account, many prefer intersections obtained by 
setting the transit over the center hub, turning 45° or other 
convenient angle from the line so that four hubs may be 
set well outside of the slope stakes with angles of 90° between 
adjacent ones. This gives two intersecting lines at right 
angles, which should give a good recovery even on rough 
ground but is not convenient if the view is cut off by the con- 


RAILROAD SURVEYING 


44 


[§ 30 . 


struction of a high fill. Approximate distances- should be 
measured to aid in finding the hubs. 

30. Track-laying and Monumenting. — In recovering 
the line for track centers after the grading is completed, some 
discrepancies occur in relocating the referenced points; 
these added to the inaccuracies of taping on rough ground 
and possibly to the inaccuracy of the instrumental work, 
usually necessitate some readjustment of alinement. Some 
shifting may also be necessary in order to fit the roadbed and 
structures. With light grading and short curves the tangents 
should be located and the Pi’s used in running the curves 
as in §12. If the Pi’s are inconvenient of access, due to 
heavy grading or long curves, the curves can best be run by 
trial from the imperfectly recovered PC’s and then corrected 
by sliding the PC’s along the tangents as in §15, converting 
any slight discrepancy in A into distance at the PT. On 
very long curves a slight compounding may sometimes be 
required to follow the roadbed. This is not especially ob¬ 
jectionable if the break in curvature is small. 

If on location care was not taken to level up the tape so as 
to bring the stations 100 feet apart, the tape should be held 
correspondingly short in running track centers so that the 
stations will coincide as nearly as may be with their old 
positions; otherwise the gradients, property lines and aline¬ 
ment will all be disturbed and confusion will result. After 
the line is in operation an entire resurvey can be made if 
desired and all these changes taken into account. 

The stakes should be 1 ^ or 2 inches square and well driven; 
they should be pointed with tacks on curves. 

The grade stakes are driven at the side with their tops at 
the height of the top of the rail, allowing for vertical curves 
at all breaks in gradients. 

The tangent points, points of change of curvature, sum¬ 
mits and, if transition curves are used, the ends of the trans¬ 
ition curves, should be marked by permanent monuments 
to guide the track foreman in maintaining alinement and for 
use in realining the track with an instrument. These should 
be set on the side at a convenient but uniform distance from 
center. Stones or pieces of old rail, from 3 to 5 feet long 
according to soil, and centered, are much used. 

It is the custom of many roads to monument also their 


PROBLEMS 


45 


right of way and other property lines for convenience and 
protection. 

Permanent BM’s should also be established, but this can 
usually be done on masonry structures or other permanent 
well-defined objects without having to place them specially 
for the purpose. 


PROBLEMS. 

1. Find the distance out to a slope stake 20.3 feet above 
grade for a 20-ft. roadbed and a 1^ to 1 side slope. 

2 . Find the width between trusses for a clearance of 7 feet 
from center of track for a bridge 90 feet long on a 3° curve. 

3 . In separating grades in a city the subgrade is raised 13 feet. 
Find the height of retaining wall required to hold the slopes for 
a double track roadbed 28 feet wide on a 40-ft. right of way. 

4 . A steel-plate girder trestle as shown in Fig. 14 is to be 
built on a 6° curve. Draw a sketch and show on it the data 
required for staking out the centers of the post piers. 



5 . An overhead highway bridge 20 feet wide is on an 8 per 
cent gradient and makes an angle of 50° with the railroad center 
line. 

(а) Find the clear span for a clearance of 14 feet for the rail¬ 
road. 

(б) Find the elevations for the tops of the piles of the pile 
abutments, using a 10 X 10 inch cap, for a clearance of 22 feet 
above center of track. Elevation of track 462.75. 

6. Find the length of 12-inch culvert pipe required under a 
20-ft. fill for a 16-ft. roadbed with 1J to 1 side slopes and a 4 per 
cent transverse surface slope. 























46 


RAILROAD SURVEYING 


7. Find the increased distance required on the inside, Prob. 2, 
to give the same car clearance if the outer rail is elevated 
3 inches; top of car body 14 feet above rail. 

8. The preliminary examination of a cut indicates that rock 
will be found at a depth of 12 feet. Find the distance out to a 
slope stake having a cut of 30 feet for a 27-ft. roadbed; rock side 
slope i to 1; earth slope 1£ to 1. 

9. If 14 feet is the minimum safe clearance in a rock cut on a 
tangent (vertical sides), find that required for a 10° curve, out¬ 
side and inside, for an elevation of outer rail of 7 inches, with a 
car body 14 feet high and 70 feet long pivoted to the trucks at 
points 48 feet apart. 

10. The abutments of a bridge make an angle of 60° with the 
axis and are 20 and 25 feet high, respectively. 

Find the distance between face lines at the base for a clear 
span of 110 feet under the coping and a face batter of one inch 
per foot. 


CHAPTER III. 


Reconnoissance. 

31. Survey for Location. — A survey for location gener¬ 
ally consists of three more or less complete surveys: — 

a. The Reconnoissance. — A rough, rapid survey, or ex¬ 
amination, of the entire belt of country within which there 
is any probability of finding a practicable route. 

b. The Preliminary Survey. — An instrumental survey or 
traverse of the line, following closely the route picked out on 
reconnoissance and serving, with its topography, as a basis 
for — 

c. The Location. — A transit and tape survey, in which 
the curves are run in and the center line is staked on the 
ground. 

These surveys, of course, generally increase in accuracy 
from the first to the third and each should vary in character 
with the local conditions. 

The problem may come to the Locating Engineer in its very 
inception as a project to build a line between two widely 
separated points, when all the broad questions of traffic, etc., 
indicated in the Introduction, must be considered in first 
selecting general routes and finally narrowing down to the 
located line. 

Usually, however, quite definite instructions are given as 
to general route, maximum gradient, degree of curve, value 
of distance, curvature, rise and fall, etc., by the higher officials ; 
who have themselves considered the larger questions involved. 
The problem is then to find the best line obtainable which will 
satisfy the requirements of the instructions. If no such line 
can be found, the Locating Engineer should report that fact 
together with suggestions as to the best procedure to over¬ 
come the difficulty. 

32 . The Reconnoissance. — The great importance of the 
location of a line was shown in the Introduction, and it is now 
easily appreciated that the reconnoissance is its most impor¬ 
tant part, as it usually fixes the general route. That is, better 

47 


48 


RAILROAD SURVEYING 


[§ 33 . 


lines are rarely found after the completion of the reconnois¬ 
sance, unless by competitors, since the selection of one route 
eliminates all others from consideration. 

Sometimes two routes will be found so nearly equal in 
desirability that a more accurate survey is necessary in order 
to decide between them, but at other times a single route can 
be picked out on reconnoissance. 

Hence the importance of employing the best possible talent 
for the survey for location and especially for the reconnoissance 
cannot be too strongly emphasized, as many of the worst errors 
in railroad location are made on the reconnoissance. 

It is also evident that the reconnoissance cannot be fully 
discussed without a knowledge of construction and operation, 
so attention will now be confined to the instruments and 
methods used and the more obvious effects of topography in 
placing the line. 

33. Types of Lines. — Flat country, with natural slopes 
not exceeding the desired maximum gradient, presents prac¬ 
tically no problem in location, the only thing necessary being 
to connect the controlling points, i.e., stream crossings, towns 
etc., by straight lines. It need hardly be said that such 
conditions are rare. 

In country approaching the above, especial care must be 
taken to avoid long, shallow cuts on account of lack of drain¬ 
age. Such country is considered somewhat difficult largely 
because it is deceptive; the apparently smooth, flat slopes 
giving, when plotted, steep gradients or heavy work which 
cannot be reduced without large lateral deviations on account 
of the flat transverse slopes. 

About the simplest of the real location problems is presented 
where a road ascends the side of a valley for the purpose of 
reaching a town or a low saddle where it can pass into the 
adjoining watershed. Rough country may, of course, m a ke 
the details of the location difficult, and the problem itself 
requires careful study if the line is to be run on ruling gradient, 
and especially if development is necessary in order to keep 
below the desired maximum gradient. 

A strictly valley line, in which the road runs along a stream 
and the advantages are alternately on one side and then the 
other, usually presents a more difficult problem. Both banks 
must be carefully examined in this case and detailed com- 


OFFICE PREPARATION 


49 


parisons made between the most promising schemes, taking 
into account not only the actual costs of the right of way, 
<iuts, fills and bridging (not only of main stream but of 
tributaries), but also the less tangible points, such as liability 
to landslides or flooding, advantages of reducing gradient, 
curvature, distance and rise and fall, and of increasing 
traffic. 

A ridge line is apt to require heavy work in order to be kept 
fairly straight, but on the whole is probably less difficult than 
a valley line. 

The most difficult general problem will be presented when 
the line cuts directly across the main drainage, since this gives 
minimum distances and maximum differences of elevation. 
High stream crossings and low saddles must be sought, but are 
rarely sufficient in themselves to give satisfactory gradients, 
and heavy summit cuts or even tunnels and “ development,” 
or purposely lengthening the line by curves, switchbacks or 
loops, are more frequently necessary in this case than in the 
others. 

In cutting diagonally across the main drainage, advantage 
may be taken of the valleys of secondary streams on one side, 
but long detours are often necessary on the ocher, though 
tertiary streams are sometimes of use, especially if fairly 
large. 

Few lines are entirely of one type and most will present a 
mixture of several different types. It is evident, however, 
that the difficulty of selecting the general route on recon- 
noissance will vary with the type of line as discussed above, 
while those attendant upon the detailed adjustment of the 
line to the ground will vary with the roughness or brokenness 
of the country. 

Hence the engineer on reconnoissance should study every 
detail of the drainage, upon which the type of line has been 
shown to depend, and not allow the ruggedness of the topog¬ 
raphy, especially if in short stretches only, to exert undue 
influence in placing the line. 

34. Office Preparation. — The first step in preparing fi>r 
the reconnoissance is to collect all available maps or other 
records. Accurate topographical maps to large scale are 
practically never available and perhaps the most useful maps 
obtainable are the small scale topographical sheets published 


50 


RAILROAD SURVEYING 


[§ 35 . 


by the United States Geological Survey, which are now avail¬ 
able over large sections of the country. The general routes 
can be selected and approximately plotted on these, even in 
fairly difficult country, thus materially reducing the labor of 
the reconnoissance. 

In laying out general routes the approximate gradients 
between controlling points are found from the differences of 
elevation, allowing for cuts and fills, if any, and the approxi¬ 
mate distances as scaled from the map. If the gradient thus 
determined does not exceed the adopted maximum, the line 
is then laid out by setting a pair of dividers at the proper dis¬ 
tance to give on that gradient a change of elevation equal to 
the contour interval and then stepping along the line from 
one contour to the next. Care must be taken to keep as 
nearly as possible to the proper position for the located line 
by jumping straight across narrow ravines and through nar¬ 
row spurs instead of following the contours exactly. The 
distance stepped off in this way should be more accurate than 
that scaled directly, and it may be necessary to revise the grade, 
when allowance may also be made for curvature if necessary. 
The method of following up this line on the ground will be 
practically the same as establishing a line, which process will 
be described later. 

Maps without contours, such as Land Office, County or 
State Maps, will be available for large territories which have 
not yet been covered by topographic surveys, but these are 
useful principally for the determination of distances. These 
outline or skeleton maps will usually show rivers and streams, 
however, and an experienced man will derive much informa¬ 
tion from a study of the drainage systems alone. 

35. Field Methods. —When no maps can be obtained and 
also when a general route as plotted is to be followed up in the 
field, means of determining distances, differences of elevation 
and direction are necessary. Since the reconnoissance is 
only a rough survey, hand or pocket instruments suffice for 
this work, though very extended exploratory surveys in new 
countries require the use of a much more elaborate outfit, 
including astronomical instruments, mercurial barometers, 
etc. Such surveys will not be treated here. See Godwin’s 
Railroad Engineer’s Field Book. 

Direction is determined by the azimuth compass. Many 


FIELD METHODS 


51 


of these are not very sensitive, and it is best to get the bearing 
of a line by taking the average of two readings obtained by 
swinging up to it from each direction. 

Approximate distances are obtained by scaling from a map, 
by timing, by the odometer or pedometer, by pacing and by 
estimation. 

The method of scaling is good for long distances when 
points can be located from the map with fair accuracy and the 
map itself is correct. No map should be accepted as correct, 
however, until thoroughly tested. The errors occasionally 
found on the Geological Survey sheets are usually such as the 
omission of roads, streams, etc., which do not affect distances 
but may confuse one as to his location. Sometimes the 
elevations are incorrect, necessitating a revision or even 
abandonment of a general route. The Locating Engineer 
should be constantly on the lookout for these errors and apply 
every possible check as he proceeds. 

Distances may be obtained within the necessary limits by 
timing, whether the method of travel be by wagon, on horse¬ 
back or on foot. This is especially true where checks can be 
obtained at long intervals, say such as may be covered in a 
day or half a day, by the use of a map, as the intermediate 
points may then be checked by interpolation. 

The odometer may be used for reconnoissance from a wagon 
and the pedometer when on foot. Reconnoissance from a 
wagon is not desirable unless the country be fairly easy, and 
the line obviously should lie near the road. Even the most 
energetic engineer, with a firm determination to investigate 
on foot all points requiring it, will often miss seeing what he 
should see, due to the use of the wagon. 

The engineer on horseback is more free to follow the lines 
unless high fences and private rights prevent. Longer dis¬ 
tances can of course be covered and with less fatigue than on 
foot. Very difficult country must be investigated on foot and 
much that is not difficult is covered in this way. Pacing 
must also be resorted to in obtaining distances, unless one 
is sufficiently experienced to estimate them closely. Timing 
and the pedometer are useless where there is much doubling 
back and forth. 

It should be carefully remembered that the more difficult 
the problem and the narrower its limits, the greater the 


52 RAILROAD SURVEYING [§ 36 . 

accuracy required. Lines are even roughly taped in some 
instances. 

The hand level is somewhat used for the determination of ele¬ 
vations, especially at critical points and in open country. The 
aneroid barometer is, however, more extensively employed, as 
it is much more rapid, can be used in wooded country and is 
accurate enough for most reconnoissance purposes if intelli¬ 
gently used. 

36. The Barometer. —This is an instrument for measur¬ 
ing the pressure of the atmosphere. It is done in the case of 
the mercurial barometer by measuring the height of the column 
of mercury which the atmosphere will support. Pure mer¬ 
cury is put into a glass tube about three feet long and one- 
quarter inch in diameter and carefully boiled to remove all 
air and moisture.* It is then inverted into the cistern of 
mercury, when the column lowers until the pressure balances 
that of the atmosphere, leaving a nearly perfect vacuum 
about one-half foot long at the top of the tube. For the 
standard mountain barometer, this tube is supported in a 
brass case which extends to the top of the cistern. The top 
of the cistern is a glass cylinder; the lower portion is of wood 
and the bottom a chamois bag. This extension is contained 
in a brass cylinder which is attached to the upper one by 
screws through outside flanges, leaving the glass portion of the 
cylinder exposed. An ivory point projects into the cistern 
from above and an adjusting screw from below supports the 
chamois bag. 

For a reading, the top of the mercury in the cistern is raised 
by the adjusting screw acting upon the chamois bottom until 
the light just disappears under the ivory point without causing 
a dimple, when the vernier tube is moved so that the plane of 
its bottom is tangent to the spherical top of the column as 
seen through two opposite slits in the brass case. 

The vernier reads to 0.002 of an inch, and the scale is ad- 

* A field method of replacing a broken tube is to fill with mercury, 
then invert, holding a piece of chamois over the end of the tube, and allow 
a small portion of the mercury to run out; the rarefied bubble remaining 
is then moved along the tube and the small air bubbles gathered up. The 
tube is refilled and the process repeated several times. A tube carefully 
filled in this way will give readings but a few hundredths lower than a 
boiled tube. 


THE BAROMETER 


53 


justable with reference to the ivory point for its zero, allowing 
for the capillary depression of the tube. 

The attached thermometer is read first, before it has been 
heated by the observer. The reading is necessary to correct 
for the expansion of the mercury and scale. 

In moving the mercurial barometer, the adjusting screw 
at the bottom should be turned up until the cistern and finally 
the tube are filled, taking care not to press hard enough to 
force a leak in the cistern. When the tube is apparently 
filled, a gentle inclination with the ear near it will indicate by 
a metallic click if it is safe to invert without further turning 
the screw. If not almost filled, there is danger of the tube's 
breaking with the blow from the mercury; if pressed too hard, 
the mercury will leak at the cistern. 

After inverting, the screw should be backed off about one- 
quarter turn to allow for the expansion with rise of tempera¬ 
ture. 

The aneroid barometer consists essentially of a box from 
which nearly all the air has been exhausted and which has its 
sides held apart by a spring which balances the atmospheric 
pressure and allows the box to expand or contract with change 
of pressure. This small motion of the sides is multiplied and 
communicated to an index moving over a graduated scale. 

The aneroid cannot be a standard barometer, as there is no 
method of graduating its scale except by comparison with a 
mercurial barometer. It is graduated in inches of mercury, 
and has an adjusting screw for shifting the zero from time to 
time, as it moves due to the wear or disturbance of the delicate 
mechanism of its parts. 

It can thus never be depended upon for actual pressures 
except when frequently compared with a standard mercurial 
barometer. It can, however, be depended upon to give cor¬ 
rect differences of pressure during short intervals of time if 
well made and carefully handled. 

Most aneroids also have an altitude scale which may be 
used for hypsometric work. It is graduated in feet to corre¬ 
spond with the pressure scale for a standard air temperature 
and its zero corresponds with standard pressure at sea level 
or other assumed datum. 

The instrument should be compensated for temperature 
which is said to be done by cutting out a portion of the side of 
one of the brass levers and replacing it with steel; differential 


54 


RAILROAD SURVEYING 


[§ 37 


expansion bending the lever to counteract the expansion of 
the other parts. 

For hints in purchasing and testing an aneroid, reference is 
made to Trans. Am. Soc. C. E., Vol. I, p. 277 (The Aneroid 
Barometer and its Use in Estimating Altitudes, by Gen. T. G. 
Ellis.) 

In using the aneroid, bring the face to a horizontal position 
and tap the case gently before reading. It should be pro¬ 
tected from jarring and sudden changes of temperature, 
hence it should be kept in the case and neither carried too close 
to the body nor exposed to direct sunlight. The index ad¬ 
justment should not be disturbed too frequently, it being 
better to apply a small index correction. 

37. Barometric Formulas. —The relation between atmos¬ 
pheric pressure and altitude is a logarithmic one on account 
of the decrease of density with altitude. Professor Church 
(Mechanics, p. 620) derives the following expression: 



(27) 


where h is the difference in elevation; p 0 ,the standard pressure, 
14.701 pounds per square inch? y 0 , the corresponding weight, 
0.08076 pounds per cubic foot, for atmospheric air; T n , the 
absolute temperature and T 0 , the absolute temperature of 
the freezing point, = 273° C.; p n and p m , the observed pressures; 
and loge, the Naperian log. 

In applying this to barometric leveling there should be 
corrections for the variation of gravity with altitude in its 
effect on the weight of the air and for the difference in den¬ 
sity due to humidity. Both of these corrections are small 
and may be omitted for reconnoissance work. Before sub¬ 
stituting the ratio of barometric readings (B n /B m ) for that 
of pressures ( p n /Pm ) all instrumental corrections must have 
been tnade. 

Mercurial barometer corrections. 

1. Reduction to standard temperatures, 32° F. for the 
mercury and 62° F. for the brass scale. 

2. Correction for the difference in the force of gravity 
acting upon the mercury in different latitudes. 


Eq. 28 .] 


BAROMETRIC FORMULAS 


55 


3. Correction for the difference in the force of gravity due 
to elevation above sea level. 

4. Correction for capillarity of the tube. 

The first correction is taken from Table VIII, which was 
computed by the following formula: 



m(t - 32°) - l(t - 62°) 
1 + m(t - 32°) 


(28) 


where C is the correction; B, the barometric reading in inches 
of mercury; m, the coefficient of expansion of mercury, = 
0.000101; t, the temperature as read from the attached 
thermometer; and l, the coefficient of expansion of the brass 
scale, = 0.000 010 2. 

This is therefore an instrumental correction for the purpose 
of eliminating the effect of the expansion of the mercury and 
the scale from the readings, and must be applied before the 
values are used for any purpose. The other corrections are 
taken care of in the formula for difference of elevation, (29). 


Aneroid barometer corrections. 

1. Correction for scale errors. 

2. Correction for temperature. 

3. Correction for index error. 


These should be determined by comparison with a standard 
mercurial barometer at different temperatures, the mercurial 
readings having been reduced to 32° F. at sea level in latitude 
45°. 

The best aneroids have no appreciable scale errors, while 
they are compensated for the temperature of the instrument. 
For an unknown instrument this should be proven by test, 
not taken for granted. 

If the same instrument is used for both readings the cor¬ 
rection for index error, unless large, may be omitted. 

There are many formulas in use for the determination of 
heights, which vary in form and somewhat in the constants 
used. The following is given in the Smithsonian Meteoro¬ 
logical Tables published in 1893 as the result of a study of the 
best available constants: 



56 


RAILROAD SURVEYING 


[§ 37 . 


Z (feet) = 62583.6 log 


Bo 

B 


[1 + 0.002039 (t - 50°)] 

(1 + /?) 

(1 + 0.002662 cos 20) (1 + 0.00239) 


(29) 


(1 + 


Z + 2 h 0 
R 


in which 62583.6 (log B 0 — og B) is an approximate value of 
Z and the factors in the parentheses are correction factors 
depending respectively on the air temperature, the humidity, 
the variation of gravity with latitude, the variation of gravity 
with altitude in its effect on the weight of mercury in the 
barometer, and the variation of gravity with altitude in its 
effect on the weight of air. 

For convenience in tabulating, log B 0 — log B is replaced 


, , 29.9 , 29.9 

by log —- !°g 


29.9 


Table IX contains values of 62583.6 log with B as 


argument. 

The table is entered with the corrected barometer reading 
for each station and the difference of tabular quantities taken 
for the approximate difference in elevation. The tempera¬ 
ture correction can be taken from Table X, or computed 
mentally by adding \ of 1 per cent for each 1° above 
50° F., or subtracting £ of 1 per cent for each 1° below 
50° F., using the average temperature for the two stations. 

The other corrections are small, and reference is made to 
the Smithsonian Tables for work in which they are needed. 

Professor Airy’s Table, which was made for graduating the 
altitude scale of aneroids, may be of value in making com¬ 
parisons. It is expressed by the following (Gen. Ellis, Trans. 
Am. Soc. C. E., Yol. I, p. 301): 


Z = 62759 



1 + 


t + t'- 100\ 
1000 / 


(30) 


where t and t f are the observed air temperatures at the two 
stations. He used 31 for B 0 , or for the zero of his altitude 
scale, which is 982 feet lower than the zero of Table IX or 
985 by the above formula, for a temperature of 50° F. 






Eq. 30.] BAROMETRIC FIELD WORK 
Formula (30) gives the following: 


57 


Inch Scale. 

Alt. Scale. 

Inch Scale. 

Alt. Scale. 

31 

0 

28 

.2774 

30.5 

443 

27.5 

3265 

30 

894 

27 

3765 

29.5 

1352 

26.5 

4275 

29 

1818 

26 

4794 

28.5 

2291 

25.5 

5323 


38. Barometric Field Work. — The above formulas 
for difference of elevation suppose simultaneous readings at 
points not far apart horizontally. However, on extensive 
government barometric surveys west of the 100th meridian, 
where sets of simultaneous mercurial barometer field readings 
were compared with the daily readings at the different 
meteorological stations, those nearest in altitude gave better 
results for elevations than those nearest in distance. This 
was in a section where the differences in elevation were 
great. 

To secure simultaneous readings, two barometers are re¬ 
quired, one at headquarters or at some fixed point and the 
other in the field. If the one at headquarters is read at the 
whole and half hours, intermediate readings can be inter¬ 
polated for comparison with the non-simultaneous field 
readings. 

The index error of the field aneroid should be determined 
each day before and after the field work and the correction 
applied before making comparison with the office record. 
Any slight change between morning and evening should be 
distributed proportionately to the time. Any large change, 
indicating a disturbance of the instrument, should exclude 
the readings unless the time of the change can be located 
and a constant value established for a sufficient time before 
and after the disturbance to warrant working up the 
results. 

Instead of determining the index error at headquarters, 
the morning and evening readings may be taken at a point 









58 


RAILROAD SURVEYING 


[§ 38 . 


of known elevation and a table of probable readings for 
this point made up by applying the constant difference to 
the office readings, this difference including index error and 
difference of altitude. 

An accurate barograph will give a continuous office record 
without the .expense of an observer. 

With a single aneroid, satisfactory results can be secured 
for short intervals of time only, on account of the erratic 
changes in the pressure of the atmosphere. For short 
intervals of time the rate of change may be considered 
constant, and approximate results obtained by reading at 
one point, then at a second, and again at the first, and 
comparing a reading interpolated for time for the first 
with the reading at the second for finding difference of ele¬ 
vation. 

Another method is to determine the rate of change of the 
barometer at as many different times as is convenient during 
the day, by taking two readings at the same point some 15 
to 30 minutes apart. These rates are then extended both 
backward and forward over about one-half the intervening 
time. 

The altitude scale should not be used except for small 
differences of elevation unless it has been compared with the 
Airy or other standard formula to see if the readings corre¬ 
spond with the pressure. If the altitude scale is movable, 
it should be set in its normal position, otherwise it will not 
give correct differences of elevation. 

An accurate altitude scale takes the place of Table IX, 
and requires only the temperature correction, which by the 
Airy formula is 0.2 per cent per degree above or below 50° F., 

+ if above, — if below, and it can be computed and applied 
mentally. 

A movable altitude scale allows of setting the zero with the 
pointer at a given station, for running gradients or for reading 
a certain small number of feet above or below the station; 
it is not intended for accurate leveling. 

If the altitude scale is used in the field, it will usually 
be less work to apply the index correction (with the sign 
changed) to the office barometer and then enter the corre¬ 
sponding altitude scale readings for comparison with the field 
readings. 


Eq. 30.] BAROMETRIC FIELD WORK 


59 


Barometric levels should always be accepted with a reser¬ 
vation unless they have been carefully checked by duplicate 
readings at important points and under varying conditions. 
Usually the results will be good, but occasionally large 
discrepancies appear which cannot be accounted for by lack 
of care or by poor weather conditions which were apparent 
at the time. 

Even the best aneroids will sometimes “ drag/’ i.e., not 
come to the proper reading at once after a considerable 
change of elevation. This necessitates a short delay at 
stations until the reading becomes constant. 


Example. — The following office readings were taken with 
a Green standard mountain barometer and a K. & E. aneroid. 


Time. 

Attached 

ther¬ 

mometer. 

Barometer. 

Aneroid. 

Observed. 

Reduced 
(T. VIIIL 

Observed. 

Correc¬ 

tion. 

7 A.M. 

60° F. 

29.642 

29.558 

29.482 

+ .076 

8 

62 

.640 

.550 



9 

65 

.635 

.537 



10 

70 

.632 

.521 



11 

72 

.624 

.508 



12 

76 

.623 

.497 



1 P.M. 

77 

.624 

.495 

29.417 

+ .078 


At 10.15 the aneroid read 29.104 with t — 75° in the field. 
Required the difference in elevation. 

Table IX, B = 29.518 350 

B 1 = 29.181 662 

Approx. Z 312 

Table X, t = 73° + 15 

Difference of elevation = 327 

The altitude scale would have given 1720 for the field reading 
and 1407 for the office reading (29.517 - . 077 = 29.440), giving 
a difference of 313 

0.1% of 46° = .046; .046 X 313 = 14 


Difference of elevation =327 




















60 


RAILROAD SURVEYING 


[§ 39 . 


39. The Preliminary Survey. — After the Locating 
Engineer has presented his report on the reconnoissance, 
giving the results briefly and including a rough estimate of 
the cost of construction, it may be decided to run a prelimi¬ 
nary survey. This will consist of a topographical survey 
of a narrow belt of country following the general route or 
routes picked out on reconnoissance and within which it is 
estimated that the final located line will lie. 

If necessary to make such a survey over more than one 
route in order to compare them, it is desirable that the line 
traversed approach very closely the trial location so that 
fairly accurate estimates of cost of construction may be 
made. Sometimes very little or no topography will be taken 
in this case and courses will be short in making swings so 
as to approximate very closely the trial location. 

Some engineers attempt to pick out courses which can be 
used for tangents in the final location even when but one 
route is to be surveyed and then take topography only at 
critical points. .Others simply survey a line close enough to 
the single general route so that the topography will surely 
cover the located line. 

Even in this latter case practice differs in respect to the 
accuracy required. Some make a very accurate survey and 
take detailed topography. The line is then plotted by 
coordinates and the topography carefully mapped. The 
location is projected, its coordinates determined and transit 
notes worked up which are then run in by the transitman 
in the field. Frequent ties to the preliminary are furnished 
him for checking his work. 

Other engineers do not attempt to make the preliminary 
especially accurate but use the projected or paper location 
simply as a guide for the final location, which they themselves 
make in the field. This keeps them constantly studying the 
country, and it is believed to be the better method. This 
question is more fully discussed in the books on railroad 
location, and is mentioned here only for the purpose of indi¬ 
cating the differences in character and purpose of preliminary 
surveys. 

In any case a traverse should be run, using the general 
field methods given in the Introduction. This is followed 
by levels taken as described in § 22. If detailed topography 


Eq. 30.] THE PRELIMINARY SURVEY 


61 


is required it is usually taken by the cross-section method, 
the instruments used being the hand level, rod and tape. 
Five-foot contours are usually required and are determined 
by selecting such a position that a rod reading on center is 
obtained which puts the eye of .the observer on a five-foot 
contour. Thus if the elevation of the station is 873.6 and the 
observer reads 1.4 on the rod, his HI is then 875. , 

The rod is then moved perpendicular to line until the read¬ 
ing is 5 feet when it is on the 870 contour and its distance 
from center is read and plotted. The rod is then moved to 
read 10 feet, and so on. In this way five-foot contours may 
be rapidly and accurately taken and plotted. 

The points where the contours cross center line should 
also be determined for accuracy in plotting and for checking 
the elevations. The distances to contours should always be 
noted as a check on the plotting and the notes should, of 
course, run up the page as in Fig. 5. 

All buildings and other structures near the line should be 
carefully located and bearings of fences, roads and streams 
determined and noted. 

The preliminary survey is sometimes made by the stadia 
method, when the topography is picked up by side shots. 
Usually the stadia will be sufficiently accurate for the purpose 
if distances and vertical angles are read on both backsight 
and frontsight, though when the line is being run on ruling 
gradient it is somewhat safer to run levels over it. The 
method is especially valuable in rugged topography since the 
salient features can be picked up and more information gained 
in a given time than by the method given above if the 
country is sufficiently open to permit the stadia shots to be 
taken. 

The organization of the party and detailed method of con¬ 
ducting the survey will vary with its character and purpose 
and the openness of the country. These subjects are fully 
discussed in various books on railroad location and will not 
be entered into here except to state that the Locating Engineer 
will take notes of all matters affecting the final location or cost 
of the line. He selects the ground ahead of the party by the 
same methods as were used on reconnoissance, supple¬ 
mented by the use of the vertical circle in running uniform 
gradients. The topographer should also take complete notes 


62 


RAILROAD SURVEYING 


[§ 40 . 


on the matters mentioned above in order to check the 
work of the engineer and insure as far as possible the collec¬ 
tion of complete data. 

40. The Location- — As soon as the map of the prelimi¬ 
nary survey or a sufficient portion of it is complete, the 
Locating Engineer starts the projection of the final location. 
If on .ruling or uniform gradient, a “ grade contour ” is very 
useful. This is plotted exactly the same as a general route 
on uniform gradient is put on the Geological Survey sheets 
(see §34), except that it must not be carried too far ahead 
of the paper location, as the corrected distances and possi¬ 
bly compensation for curvature will necessitate a revision of 
the grade elevations. This grade contour is, of course, a 
surface line, i.e., a line of no cut or fill along the center 
line, and it will assist greatly in properly placing the trial 
location. 

Profiles of trial projections are easily made up by noting 
where they cross the contours, and from these, or directly from 
the maps, quantities may be determined and costs estimated. 
Costs of structures on alternate projects need not be deter¬ 
mined for comparisons unless they are so different as to require 
different structures. 

The detailed comparison of the trial locations must include 
many factors aside from cost of construction. These factors 
involve a knowledge of the entire field of railroad engineering 
and hence cannot be discussed here. 

As fast as the projected location is finished, the center line 
is staked on the ground as before described. Usually the 
transition curves, if used, are not run in, but offsets are 
allowed for them. 

In staking the center line, property lines, roads, streams, 
etc., are located and right of way maps made up and filed as 
may be required. Deeds for right of way are obtained and the 
line is made ready for construction as described in Chapter II. 


Eq. 30.] 


PROBLEMS 


63 


PROBLEMS. 

1. A T. & S. aneroid barometer has a movable altitude scale 
which gives the following readings for different settings of the 
scale: 


Inches. 

Feet. 

Feet. 

Feet. 

Feet. 

31 

0 

190 

380 

570 

30.5 

445 

637 

825 

1016 

30 

891 

1081 

1276 

1471 

29,5 

1341 

1537 

1733 

1929 

29. 

1800 

1995 

2199 

2406 

28.5 

2270 

2476 

2680 

2888 

28 

2751 

2958 




Show which of these settings would give results corresponding 
most closely with (30). 

2. On the day the mercurial barometer readings of § 38 were 
taken, an aneroid was read at Sta. A some 400 feet lower and 
several miles distant as follows: At 8 a.m., 904 ft.; 1 p.m., 960 ft. 

Find the height of Sta. B below Sta. A for a reading at Sta. B 
of 460 ft. at 9.45 a.m. with an air temperature of 71° F. 

3. Obtain a Geological Survey sheet on which are two towns 
on the same side of a valley at least eight miles apart and plot a 
general route for a railroad on a uniform gradient. 

4. An engineer having only one aneroid barometer took the 
following readings for rates of change. 

8 a.m. 29.301 12 m. 29.407 5 p.m. 29.612 

8.15 a.m. 29.309 12.30 p.m. 29.403 5.20 p.m. 29.613 


With times for abscissas draw a curve of probable corrections. 













CHAPTER /IV. 


Turnouts. 

41. General. —- A turnout is a track for passing from the 
main track to a siding or other track. It consists of three 
parts as shown in Fig. 15; the movable switch or switch rails 
AD and BE for turning from the main track, the lead rails 



Fig. 15. 


KF and GP for leading to the frog, and the frog at F to 
provide open flangeways for wheels to pass the intersection on 
either track. 

The movable ends of the switch rails, which may be at A 
and B or at D and E, are controlled from a switch stand 
placed outside the track on a long tie called the head-block. 
The distance from the head-block to the point of frog, 
measured along the main track, is called the lead; stub lead 
if the head block is at D as for the stub switch, and point lead 
if at A as for the point switch. 

The point of switch is at A, which is the heel of the stub 
switch rail and the toe or point of the point switch rail. 

The throw is the distance the toe of the switch rail is moved 
in opening or closing the switch. The spread is the distance 

64 






Eq. 31.] 


STUB SWITCH TURNOUT 


65 


between main and switch rail centers at the heel of the 
point switch. Some 5 to 5f inches is required to give 
clearance for wheel flanges. 

The gage of a track is the width between the rail heads 
measured at points -| of an inch below the tops of the rails. 

The ordinary turnout frogs are numbered, the number 
being the axial length from the theoretical point, or inter¬ 
section of the sides produced, to where the width is unity. 
Since this length is one-half the cotangent of the frog angle F, 
the frog number, 

n= ^ cot ^ F. (31) 


42. Stub Switch Turnout from Straight Track. — In 

the stub switch, Fig. 15, the ends of the switch rails back of A 
and B are spiked down so that in throwing the free ends to 
K and G for the turnout they are bent to a curve. On this 
account it is customary to assume a circular arc for the 
turnout curve, although the frog is usually straight except 
*for street railroad track. 

Given the frog angle F, the throw t, and the gage g ; to 
find the radius of the turnout, r, the point lead and the length 
of the switch rail l. 

In the triangles AFO' and ABF , 


r + - q = -— • 

2 vers F 

Point lead, BF = (r + ^ g) sin F, > 
or, = g cot \ F. ) 


(32) 

(33) 


Similarly in the triangle ADO' (neglecting the effect of 
gage), 

Vers AO'D = 

r 

1= r tan AO'D. (34) 


Since for small angles tangent offsets increase approxi¬ 
mately as the squares of the tangents, 



66 RAILROAD SURVEYING [§ 43 . 


These values for laying out the turnout are usually given 
in terms of the frog number n. 

Thus from (31) and (33), 

BF = 2 gn. (36) 


From the triangle BFO', 

(FO') 2 = (BF) 2 + (BO') 2 , or 
(r + J g) 2 = (2 gn) 2 + (r - \ g) 2 . 

Solving, 


r = 2 gn 2 . 

From (35), (36) and (37), 


l = 2n\/gt=\/2 rt. 


(37) 

(38) 


For the standard gage of 4 feet inches and a throw of 
5 inches, these formulas become 

Lead, BF = 9.42 n, 1 

Radius, r = 9.42 n 2 , >* (39)* 

Switch rail, l =2.8 n. ) 

The lead rail may be laid out by ordinates from the main 
rail AI proportional to the squares of the distances from A, 
giving at the quarter, half and three-quarter points, y 1 ^ g, 
l g, and g, respectively. Or, the middle ordinate of the 
chord AF = (r + \ g) vers \F, and the side or quarter 
ordinates will be three-quarters the middle ordinate. 

The circular arc assumed in developing these formulas was 
approximately secured when cast-iron frogs were in use. 
At present much longer frogs are made up from rail sections, 
and usually without curvature in steam railroad practice. 
Again, 22 feet is about the maximum allowable distance for 
an unspiked switch rail, and this corresponds to about a 
No. 8 frog; so that for large frog numbers the switch rail 
will not coincide with the curve, while for small ones the 
frogs will not coincide unless the leg is bent. 

On the other hand, variations of 10 per cent are frequently 
made in the lead to avoid cutting a rail or for other reasons, 
with but slight effect upon the riding qualities of the turnout 
43. Stub Switch Turnout from Curved Track. — The 
lead and length of switch rail are practically unaffected for 


Eq. 40.] 


POINT SWITCH TURNOUT 


67 


curved main track until the curvature becomes quite sharp. 
The degree of curve is increased or decreased by that of main 
track according as the turnout is on the inside or on the out¬ 
side of the curve, and the lead rail ordinates can be used as in 
§42. On this account and because the stub switch has prac¬ 
tically gone out of use, at least for main line, the derivation 
of the accurate formulas is omitted. 

44. Stub Switch Turnout with Tangent at Frog. — 

The effect of the tangent l' at the frog F, Fig. 16, so far as the 



Fig. 16. 


turnout up to the tangent point C is concerned will be to 
replace the gage by (g— V sin F). 

Then from (36), (37) and (38), 

Lead, BF = 2 n(g — V sin F) + V cos F, 

Radius, r =2 n 2 (g — V sin F), C (40) 

Switch rail, 1= 2 n\/ t(g—l r sin F) = \/2rt. ) 

The effect is to shorten the lead by the length of the tan¬ 
gent, i.e., A A' = l f , and to sharpen the curve. The frog is 
usually laid out straight, as already indicated. 

45. Point Switch Turnout from Straight Track. — In 
the point switch, the outside rail of the main track and the 
inside rail of the turnout are continuous past the switch rails, 
a slight kink in each protecting the sharp points of the straight 
(on the gage side) switch rails. This gives a continuous rail 
support for the wheels for either main line or turnout, and 






68 


RAILROAD SURVEYING 


[§ 45 . 


prevents the jolting at the toe, so characteristic of the stub 
switch. It is also less sensitive to relative movement of 
rails due to creeping or temperature changes. For these 
reasons it has practically replaced the stub switch, although 
rather more sensitive to disturbance by ice and snow. 

In Fig. 17, AK is the straight switch rail in position for 
the turnout curve KF and making the angle S, called the 



Fig. 17. 


switch angle, with the main track. The long chord KF 
makes an angle = ^ (F — S) with the switch rail AK, or 
J (F + S) with the main rail. 

In the triangle DQF, 

9 ~ t 


But 


K F 1 — 

sin i (F + S) ' 

1 KF 

r 4- - a — - y or 

2 9 2 sin | (F - S) . 

1 9 - t 

T + 2 9 2 sin 1 (F + S) sin £ (F - S) 

9 ~ t 


cos S — cos F 


(41) 


The last value is found by substituting for 2 sin (F + S) 
sin 4 (F — S ) its equal (cos S — cos F), and will be conven- 











Eq. 42.] 


POINT SWITCH TURNOUT 


69 


ient for field computations with natural functions, while the 
first form is better for use with logarithms. 

t 

sin S = ~ - 

L 

BF = l cos S + KF cos ^ (F + S), or 

Point lead, BF — l cos S + (g — t) cot ^ (F + S). (42) 

With a tangent V at F the gage would be replaced as 
before, giving 

1 g — t — V sin F 

r + = 2 sin l (F + S) sin \ (F - S) 

- g ~ * ~ l - 8iD F , (43) 

cos S — cos F 

BF = l cos S + (g — t — l' sin F) 
cot (F S') -j- V cos F. 

The middle ordinate for the chord from the heel K to the 
frog, point or toe as the case may be, will be 

m — (r + J g) vers J (F — S). (44) 

46. Point Switch Turnout on the Inside of Curved 
Track. — With curved main track the* gage lines of the 
switch rails have the curvature of the track, so that the 
turnout at the heel of the outer switch rail makes the switch 
angle S with the main track the same as for straight track. 

The center of the turnout will thus be at O', Fig. 18, on the 
line KO', making the switch angle S with the radius KO. 
For convenience use the radius r of the outer rail of the turn¬ 
out and R' of the inner rail of the main track. 

In the triangle KOO', KO' = r; KO = R' + a, where 
a = g — t; angle 0K0' = S; 00' = b. 

By trigonometry, 

b 2 = r 2 + ( R' + a) 2 — 2 r ( R' + a) cos S. 

In the triangle F00', F0'= r; FO = R'; angle OFO' => 
frog angle F, 

b 2 = r 2 + R' 2 — 2 rR' cos F. 

Placing the two values of b 2 equal, 

R' 2 + 2 aR’ + a 2 - 2 r (R' + a) cos S = R' 2 - 2 rR' cos F. 





70 


RAILROAD SURVEYING 


[§ 46 . 


Radiul of outer rail, r = 


aR' + ^ a 2 


-(45) 

(R' + a) cos S - R' cos F 
For the angles at 0 and O' in the two triangles use tan % (A — B) 
a — b 

= -- tan \ (A + B). 

a -f- b 


D 



The difference of those at 0 will give the central angle for the 
stub lead, and of those at O' that for the length of the turnout 
from heel to frog. 


Example. — Find the data for laying out a turnout with a 
no. 10 frog, F = 5° 43F, on the inside of a 4° curve; 18-ft. switch 
rail, in. spread, giving S = 1° 27^'. 

R 4 = 1432.5 ,R' = 1430.15;a = 4.71 - 0.46 = 4.25. 


Solution. 
(45), r = 


4.25 X 1430.15+ i (4.25) 2 


(1430.15 + 4. 25) 0.99968 - 1430.15X0.99501 
6087.1 


10.94 


= 556.40. 


Triangle FOO'. 


FO - FO' = 1430.15 - 556.40 = 873.75 ... 2.94138 

FO+ FO' = 1430.15+ 556.40 = 1986.55+4(7)6.70190 

%F = 2° 52', cot. 1.30038 


= 83° 30', tan. 

0/ = 170° 38'; 0 1 = 3° 38'. 


* W - O t ) 


0.94366 










Eq. 46.] point switoh turnout 71 

Triangle KOO'. 

KO - KO' = 1434.40 - 556.40 = 878.00 . 2.94349 

KO + KO' = 1434.40 + 556.40 = 1990.80(^0)6.70097 

4 S = 0° 44', cot.. 1.89280 

£ (0 2 ' + 0 2 ) = 88° 20', tan. 1 . 53726 

0 2 ' = 177° 36'; 0 2 = 0° 56'. 


KOF = O, - 0 2 = 2° 42'; stub lead EF = 162'/240i' = 
67.4 ft. 

KO'F = 0 2 ' - 0/ = 6° 58'; KF = 418'/618' = 67.6 ft. 

47. Point Switch Turnout on the Outside of Curved 
Track. — For convenience use the radius r of the outer rail 
of the turnout and R" of the outer rail of the main track, 
with a = g — t. 



In the triangle KOO', Fig. 19, as in § 46, 

6 2 = r 2 + ( R" - a) 2 + 2 r ( R" - a) cos S. 

In the triangle FOO' 

b 2 = r 2 + R" 2 + 2 rR" cos F. 

Equating, 

- 2 aR" 4- a 2 + 2 r (72" - a) cos £ = 2 rR" cos F. 
aiT - \ a 2 

(iT - a) cos S - R" cos F ’ 


r 


(46) 








72 RAILROAD SURVEYING [§ 48 . 


Having r, the angles at 0 and O' are found as in the pre¬ 
ceding section. 


Example. — Find the data for laying out a turnout with 
a no. 6 frog, F = 9° 31+, on the outside of an 8° curve; 15-ft. 
switch rail, 5* inch spread, giving S = 1°45'; F s = 716.25; 
R" =718.60; a = 4.71 — 0.46 = 4.25. 


Sohition. 


(46), 


_4,25 X 718.60 - * (4.25) 2 _ 

(718.60 - 4.25) 0.99953 — 718.60 X 0.98622 


Triangle FOO'. 
FO - FO' 

FO+ FO' 
i ( 0 / + O,) 


3045.05 

5.32 


572.37. 


= 718.60 - 572.37 = 146.23. 2.16504 

= 718.60+ 572.37 = 1290.97 (AC) 6.88908 
= 4° 46', tan. 8.92110 


* (0/ - OJ = 0° 32+, tan. 7.97522 

O / = 5° 18+; 0 1 = 4° 13+. 

Triangle KOO '. 

KO - KO' = 714.35- 572.37 = 141.98. 2.15223 

KO+ KO' = 714.35+ 572.37 = 1286.72 (AC) 6.89052 

i (0/ + 0 2 ) = 0° 52+, tan. 8.18390 


£ (0/ - 0 2 ) = 0° 6', tan. 7.22665 

0 2 ' = 0°58+ ;0 =0° 46+. 

KOF = Oj - 0 2 = 3° 27'; stub lead EF = 207/478* =43.2 ft. 
KO'F = 0/ - 0 2 ' = 4° 20'; lead rail KF = 260/601 =43.3 ft. 

48. P oint Switch Turnouts by Tables— Table XI, from 
Camp’s Notes on Track, gives the data for laying out turn¬ 
outs from straight main track without computation. 

For curved main track he finds that the lead should be 
increased, or decreased, by Dn 2 /144, according as the turn¬ 
out is from the inside or from the outside of the curve. D is 
the degree of the main track curve and n is the frog number. 

For the middle and quarter point ordinates from the outer 
rail of the turnout curve to the chord joining the heel with 
the point of frog the ordinates for straight main track are 
divided by the degree of turnout curve for straight track and 
multiplied by the degree of curve for main track. One and a 
quarter times these values are added if the turnout is on the 
inside, or one and an eighth times are subtracted if the turn¬ 
out is on the outside. 

Within the ordinary range of frog numbers and degrees of 












Eq. 47.] TURNOUTS WITHOUT instruments 


73 


curve it is claimed that these corrections will give results 
checking the true values quite closely. 

49. Three Throw Point Switch Turnout. — If the main 
line frogs are opposite and the turnouts symmetrical, as 



assumed in Table XI and shown in Fig. 20, ^ g and \F" would 
replace g and F in the second members of (41) and (42), giving 
Point lead to crotch frog, 


BH = IcosS + (Jgr - 0 cot £ (JF" +5), 

cos 4 F"= cos S — • 

2 r+ ig 


(47) 


If the main line frogs are not opposite, or if the turnouts 
are not symmetrical, the data given would locate the centers 
O' and 0", and the problem would be a crossing frog problem 
which could be worked in a manner similar to those of §45 
and §46. 

Thus if O' were moved to 0/, the distances 0/ J , JH r and 
H' 0" would be known, locating the centers. 

50. Turnouts without Instruments, Formulas or 
Tables. — In making out a set of turnout standards and in 
many other cases it will be desirable to make use of the 
formulas and table given for laying out turnouts. Some¬ 
times, however, the following graphic methods for making a 
direct solution on the track will be found helpful. 






74 


RAILROAD SURVEYING 


[§ 50 . 


Thus for the stub switch of Fig. 15, lay off 4 n + — (Eq. 1.) 

2 n 

on the gage line of the rail to the left from F, then locate a 

point 4 feet from this and 4 n + — from F and join with F 

2 n 

for the frog tangent. Produce this line by tape or by eye 
to I for the PI of the turnout curve. 

Lay off IF on the main rail to locate the heel at A. Draw 
the chord AF and drop the perpendicular IJ . This per¬ 
pendicular is bisected by the turnout curve, and the side 
ordinates from the chord will be three-fourths of the middle 
ordinate. 

A tangent V at the frog, as in Fig. 16, will change the outside 
lead rail chord to A'C. 

For the point switch, Fig. 17, after laying out the frog 
tangent as before, estimate the distance from I to the main 
rail and measure FI; multiply (FI -(- l ) by the spread of 
the point rail per foot (t/l) and compare with LI. A second 
trial should suffice, as the frog angle is greater than the switch 
angle and an exact equality of the tangents is not essential. 

If the track is curved at the frog, measure 4 n + — 

2 n 

each way from the frog point, draw the chord and measure 
the middle ordinate; add or subtract this ordinate or tangent 
offset as the case may be, in laying off the frog tangent IF. 
The same method can be used in laying off the tangent to 
a curved switch rail. 

If for any reason the lead for a circular curve is changed, 
making the tangents unequal, a parabola connecting these 
unequal tangents can be laid out by joining the vertex with 
the center of the chord and drawing the side ordinates parallel 
with this center one; the lengths being obtained as in Fig. 15 
for the circle. Or, the length of the shorter tangent can be 
laid off on the longer, the circular arc drawn, and the excess 
left as a tangent to the circle. 

For complicated turnout and crossing problems these field 
methods can be adapted to the draughting room, using a 
scale of from 5 to 10 feet per inch and checking the relation 
between distance, central angle and curvature by compu¬ 
tation for increased accuracy. 


Eq. 47.] 


PROBLEMS 


75 


PROBLEMS. 

1. Find the lead and radius for a 5-ft. tangent at the frog of a 
point switch turnout for a no. 8 frog. 

2 . Find the lead and radius for a 5-ft. tangent at the frog of a 
point switch turnout on the outside of a 4° curvp for a no. 7 frog. 

3 . Find the data for staking out a crossover between two 
straight tracks, 12 ft. centers, for stub switches, no. 10 frogs, the 
turnout curves being continued beyond the frogs to a reversing 
point. 

4 . Two turnouts on double track, 13 ft. centers, are connected 
by a tangent between frogs so as to form a crossover from one 
track to the other. 

Find the data for laying out with no. 11 frogs and 20-ft. point 
switch rails with 6-in. spreads. 

5. On location a 4° curve intersects the first rail of a single 
track tangent at a deflection angle of 28° 32'. 

Find the frog angles and the distances between frog points on 
both tracks for a single track grade crossing. 

6. Compare the radius and lead of a stub switch turnout for 
a no. 9 frog with those of a point switch having a 15-ft. switch 
rail and a 5^-in. spread. 

7 . Check the rule of §48 for lead and middle ordinate for a 
point switch turnout on the inside of a 3° curve by comparison 
with the values computed by §46; no. 11 frog; 18-ft. switch rail, 
and 5^-in. throw. 

8. Find the crotch frog angle and lead for a 10-ft. tandem three 
throw point switch turnout with no. 9 frogs and 20-ft. switch 
rails. 

9 . Find the middle and side ordinates from the turnout chord 
AF for a stub switch turnout on the inside of a 3° curve, using a 
no. 10 frog. 

10 . For a stub switch turnout with a no. 12 frog, compare the 
theoretical lead and radius with the actual, if the lead rail curve 
is tangent to the switch rail at the toe and the free portion of the 
switch rail is an arc of a circle 22 ft. long. 


CHAPTER V. 


Special Problems. 

51. Middle Ordinates for curving Rails. — In Fig. 21, 
let C be the center; AD, a diameter and FE, or c, a chord of 
the simple curve F AE whose middle ordinate is AB, or m. 



In the right triangles ABE and ADE, the angles AEB and 
ADE are equal, being measured by one-half the equal arcs AF 
and AE. Therefore the triangles are similar and 

AB : AE = AE : AD. 


But for flat arcs AE is practically \ FE, or % c. 
ing and reducing, 

c 2 c 2 D 

m = —— = - (approx.). 

8 R 45 840 v ’ 


Substitut¬ 


es) 


The use of this formula may be illustrated by determining 
the length of the chord whose middle ordinate in inches equals 

the degree of the curve. Since rri in the formula is in feet, — 

' 12 

will be its value in the assumed problem. 


76 





Eq. 49.] ELEVATION OF OUTER RAIL ON CURVES 77 


Substituting this value for m 

c = 61.8 feet. 

That is, to find the degree of a curve approximately, deter¬ 
mine the length in inches of the middle ordinate for a chord 
of 61.8 feet. This fact is useful in determining the degree of 
curve on old track, and the formula is also used £o find the 
middle ordinate for curving rails to a given radius. Table 
XII was computed by its use. 

It is also used to determine the length of chord whose middle 
ordinate equals the desired elevation of outer rail on curves, 
say f inch per degree of curve, etc. 

52. Correction for Curvature and Refraction in 
Leveling. —In applying § 51 to this problem, the radius is the 
mean radius of the earth, r (= 20 890 129 feet), and the length 
of sight, L, is one-half the chord c, giving 


. L 2 L 2 

Correction = — = - 

2 r 41 780 258 


(49) 


The effect of refraction is to reduce this, since it makes the 
line of sight slightly concave downward. The radius of the 
line of sight varies, due to variations in refraction, but aver¬ 
ages about 7 r, giving for the correction 


c> _ IE = L2 . 
7 r 48 743 634 


(50) 


This correction is subtractive from a rod reading or additive 
to elevation. It amounts to 0.020 feet in 1000 and varies as 
the square of the distance. 

The correction should be applied when a long frontsight is 
necessary, as in crossing a stream, but the results should not be 
depended upon for accurate work without checking by a back¬ 
sight on account of the uncertainty in refraction. 

53. Elevation of Outer Rail on Curves. — The outer 
rail on curves is made higher than the inner one, so that the 
weight will have a component towards the center of the curve 
which will tend to balance the centrifugal force. From 
Mechanics, — 

Mv 2 

Centrifugal force == ——- > 


( 51 ) 




78 RAILROAD SURVEYING » [§ 53 . 


where M is the mass; v, velocity and R, radius of curvature, 
all in the foot-pound-second system. 

And on an inclined plane making an angle a with the 
horizontal, 

Component of weight along plane = G sin a, (52) 
where G is weight in pounds. 

Let e be, the elevation of outer rail and g' the distance be¬ 
tween centers of rails, both in feet. 


Then 


e 

— = sin a, 
9 


also 


M = 

v 2 = 


G 

32.16’ 

V 2 (5280) 2 
(3600) 2 ’ 


where V is in miles per hour. 

Equating the expressions for the two forces for equilib¬ 
rium, substituting the above values and solving, 


0.06686 yy 

e =---- = 0.000 011 7 V 2 g'D. (53) 


The distance g' is slightly greater than the gage of the track, 
but since the rail can be elevated for only one speed, which 
may never be the exact speed of the train, nothing is gained 
by splitting hairs and the gage will be used for g'. 

For the standard gage of 4 feet 8£ inches (53) becomes 

0.315 V 2 

e = --— = 0.000 055 V 2 D. (54) 


For the elevation in inches (E = 12 e), 

„ 3.78 P 

E = --—- = 0.000 66 V 2 D. (55) 


Table XIII, from the Manual of Recommended Practice of 
the Am. Ry. Eng. and M. of W. Assoc.,,was computed by this 
formula. 

When tracks are used for only one kind of service, so that 
the speed on a given curve is nearly constant, it is possible 
to apply the proper elevation for that speed. In the usual 







Eq. 55.] 


VERTICAL CURVES 


79 


case, with varying speeds, it is customary to elevate for nearly 
the highest one, or for fast passenger trains, on account of 
safety. With the centrifugal force balanced there is still a 
flange pressure on the front outer wheels of the trucks due to 
the use of parallel axles. 

Equation (55) may be used with (48) for finding the chord 
whose middle ordinate in inches will equal the proper eleva¬ 
tion of outer rail, for the convenience of the trackmen. 

The Manual mentioned above also gives the following 
recommendations: — 

“ Ordinarily an elevation of 8 inches should not be exceeded 
and the speed of trains should be regulated to conform to that 
elevation. 

“ The elevation of curves should be zero at the point of 
spiral (or transition curve) and should increase to full eleva¬ 
tion at the end of the spiral or beginning of the simple curve. 

“ In ordinary practice it is recommended that the ele¬ 
vation be run out at the rate of 1 inch in 60 feet, but this will 
be modified by the same conditions that should vary the length 
of the easement curve used. 

“ The inner rail should be maintained at grade.” 

54. Vertical Curves. — Vertical curves are necessary at 
changes of gradient in order to avoid undue shock or slacken¬ 
ing of the couplings. 

Theoretically, their length should depend upon the total 
change of gradient and the length of the longest train. Prac¬ 
tically, the length is usually found by dividing the total change 
of gradient by the allowable change per station, though some 
roads use an arbitrary length, say 400 feet, for all cases. 

The Am. Ry. Eng. and M. of W. Assoc, recommends a rate of 
change of 0.1 per cent per station on summits and 0.05 per cent 
per station in sags for first-class lines and double these values 
for minor lines. 

The gradient of the curve at any point is that of its tangent 
at that point. For a station length, the average gradient is 
that of the tangent at its middle point, or its chord which is 
parallel to that tangent. Hence the change of gradient per 
station is the chord offset of the curve. That is, suppose the 
chords for two consecutive stations have gradients of + 0.4 per 
cent and +0.6 per cent, respectively. The rate of change of 
gradient is 0.2, and if the chord for the first station length be 


80 


RAILROAD SURVEYING 


[§ 54 . 

extended over the second the chord offset will be 0.2. The 
tangent offset per station will be 0.1, since, as shown in Chapter 
I, the tangent offset is one-half the chord offset. 

The degree, or radius, of a vertical curve is never found, as 
they are easily laid out by the tangent offset method, using the 
fact developed in the last paragraph, i.e., that the tangent 
offset per station is one-half the rate of change of gradient. 



Fig. 22 shows the general case of a vertical curve beginning 
and ending between stations. The two grade lines, EG and 
GJ, meeting at G are connected by the vertical curve ADIB. 
The beginning of the curve, A, lies between the regular stations 
C and E. Call its distance from C, n, in stations. AE will 
then be 1 — n. Extend the curve backward to F. 

Let t be the tangent offset per station (= \ the change of 
gradient), then 

DC = n 2 1 and 
' FE = (1 — n) 2 1. 

The difference -between these two offsets applied to the 
gradient oi EG will give the gradient of FD which is one 
chord of the curve. The gradient of the succeeding chords 
can then be found in turn by applying the rate of change of 
gradient per station. 

To find the grade elevation of D, first find that of C and add 
DC. This gives a starting point for the grade elevations of 
the regular stations, and having the gradients of the chords 
the others are readily computed. A similar process serves 
to get from curve to tangent at the other end, and the grade 
of B should always be figured both along the curve and along 
the tangents, thus checking the computations. 


Eq. 55.] 


VERTICAL CURVES 


81 


Example. — A — 0.6 per cent gradient and a + 0.4 per cent 
intersect at sta. 186+ 30 with a grade elevation of 217.35. 
Determine grade elevations at the regular stations on a vertical 
curve changing gradient at the rate of 0.2 per cent per station. 

Solution. — Total change of gradient = + 0.4 — (— 0.6) = 
1.0 per cent. 

Length of curve = ^ = 5 stations. 

PC = 186 + 30 - (2 + 50) = 183 + 80. 

PT = 186+ 30 + (2+ 50) = 188+ 80. 

n = .20; 1 — n— .80. 

Grade of PC = 217.35 + 2* X 0. 6 = 218.85. 

Grade of PT = 217.35+ 2* X 0.4 = 218.35. 

Offset at 184, 0.2 2 X 0.1 = 0.004 
Offset at 183, 0.8 2 X 0.1 = 0.064 

Difference . =0.06 

Gradient of chord FD = — 0.66 per cent. 

Grade of 184 on tangent, 218.85 — .2 X 0.6 = 218.73. 
Grade of 184 on curve, 218.73 + .004 = 218.734. 


Sta. 

Gradient. 

Grade. 

183 + 80, PC 

184 

185 

186 

187 

188 

+ 80, PT 

-0.46 
-0.26 
-0.06 
+ 0.14 

218.85 

218.734 

218.274 

218.014 

217.954 

218.094 

218.35 


w 188 + 80 PT, 218.094 - .064 + 0.8 X 0.4 = 218.35, checking 
the above value. 


Vertical curves can usually be made to begin and end at 
regular stations, when the above method will be more easily 
applied, since the offsets need not be computed. That is, 
simply change the gradient of the first chord by one-half the 
rate of change per station and the others by the full rate. 
This should give a grade checking the PT, and the gradient 
of the last chord should differ from that of the tangent by 
one-half the rate of change per station. 










82 


RAILROAD SURVEYING 


[§ 55 . 


For the general case most engineers prefer another method 
in which the tangent offsets from the first tangent for the 
entire curve are figured by making them proportional to the 
squares of the distances from the PC. This method will be 
applied to the example given above. 


Sta. 

Grade on 
Tang. 

t'=n 2 t. 

Grade on 
Curve. 

183 + 80, PC 

218.85 


218.85 

184 

218.73 

.004 

218.734 

185 

218.13 

.144 

218.274 

186 

217.53 

.484 

218.014 

187 

216.93 

1.024 

217.954 

188 

216.33 

1.764 

218.094 

+ 80, PT 

215.85 

2.50 

218.35 


55. The Stadia Method. — This will be found so useful in 
railroad work that it should be reviewed and the following 
formulas derived: 

D = kr cos 2 a + (/ + c) cos a, (56) 

H = kr cos a sin a + (/ + c) sin a, (57) 

where D is the horizontal distance; H, the difference of ele¬ 
vation; k, the stadia constant; r, the rod reading or intercept; 
a, the angle of inclination;/, the focal length of the objective 
and c, the distance from the center of the objective to the 
center of the instrument. 

For horizontal sights (56) becomes 

D = kr + f + c. (58) 

Subtracting (56) from (58), 

Correction = kr (1 — cos 2 a) + (/ +c) (1 — cos a) 

— kr sin 2 a (nearly). 

This correction, subtracted from the distance found by 
neglecting the slope, will give the true distance as accurately 
with three places in the computation as the direct method 
will with five up to 6°, or with four up to 18°. 

The vertical distance is the altitude of the triangle of base 

D, i.e. y 



H = D tan a. 


( 60 ) 










Eq. 60.] 


TRANSIT ADJUSTMENTS 


83 


A diagram made up as described in the Jour. W. Soc. of 
Eng., Vol. Ill, p. 1399, will give both horizontal and vertical 
distance with one entry. For occasional field use, Table XIV, 
giving corrections to horizontal distances computed by (59), 
and natural tangents for differences of elevation, will be found 
more portable and convenient. 

For accurate work the constant k should be determined by 
taking readings for various measured distances on level ground 
and solving for k in (58), / and c being measured on the 
instrument. In using Table XIV, multiply the correction 
from the table by the rod reading, or intercept, in feet, cor¬ 
rected for constant if necessary, and subtract the result 
from 100 times the (corrected) rod reading. For vertical 
angles over 18° subtract 0.1 ft. from the final result for 
(/ + c) (1 — cos a) in computing to tenths of feet. 

56. Adjustment of Instruments. — The adjustments of 
the instruments should be taken up in the order given below. 
Before using an instrument, however, and especially before 
attempting to adjust it, the observer should focus the eye¬ 
piece upon the wires. Otherwise, errors due to parallax will 
enter into the work and make a good adjustment impossible. 

This is done by focusing the eyepiece sharply on the wires 
while the field is made clear of any image by pointing at the 
sky or a piece of blank paper or even by throwing the objec¬ 
tive out of focus. The focusing should be tested by sighting 
upon some object and noting if there be any relative motion of 
wires and image as the eye is moved and repeated if necessary. 

In correcting the errors it is best to under-adjust rather 
than over-adjust, and care should always be taken that the 
screws have an even and firm bearing. In using two oppos¬ 
ing screws, one should always be slightly loosened before the 
other is tightened. In adjusting the line of collimation care 
must be taken not to disturb the verticality of the vertical 
wire. In putting in the reticule, or in any case where the 
adjustments are badly in error, they must be gone through 
with roughly before attempting the final adjustment. 

a. Transit Adjustments. 

1. Plate Levels. — Level carefully, being sure that the 
levels are parallel with the leveling screws. Swing 180° in 
azimuth, thus reversing the levels and doubling their errors. 


84 


RAILROAD SURVEYING 


[§ 56 . 


Correct one-half the apparent error by the use of the adjust¬ 
ing screws, bring bubbles to the center by the leveling screws 
and repeat. 

It is best to work with one level at a time, beginning with 
the one having the larger error. Also rotate by use of the 
upper motion, as it is more important to have the levels 
correct for this motion if there is any difference between 
the two. 

2. Vertical Wire. — If this is truly vertical it will cover 
a point upon which the transit has been set as the telescope is 
moved slightly up and down. Test in this way and correct 
if necessary by loosening the pairs of screws holding the 
reticule and turning it until the wire stands the test. 

3. Line of Collimation. — Set up at about the middle of a 
level stretch at least 400 feet long. Backsight on a well- 
defined point and, plunging, set a point ahead. Swing in 
azimuth to the backsight. Plunge again, and correct one- 
fourth the apparent error, if any, by moving the vertical wire. 
Repeat the test and correction until the same point is obtained 
on frontsight whether the backsight is taken with the tele¬ 
scope direct or reversed. 

4. Horizontal Axis. — Pick out a well-defined, high point 
on a bpilding and set up so as to give as large a vertical angle 
as convenient for use. Drop down and line in a point near the 
ground. Swing 180° in azimuth, plunge and repeat. If the 
two lower points do not coincide, adjust the movable end of 
the telescope axis by trial until a point half-way between 
them is obtained in dropping down from the upper point. 

5. Telescope Level. —This is made parallel to the line of 
collimation by the “ two-peg ” method. Set up half-way, by 
pacing, between two pegs or other solid points and read a rod 
on each with the bubble centered in each case. The differ¬ 
ence of rod readings gives the true difference of elevation even 
if the level is not in adjustment, since the inclination of the 
line of sight would be the same in both cases and, with equal 
distances, would give equal errors which neutralize each other. 

Now set up near one of the points and take a reading upon 
it. Apply the difference of elevation to this reading, thus 
obtaining a correct reading for the far point. If this read¬ 
ing is not obtained with the bubble centered, set the wire upon 
it and bring the bubble to the center by the adjusting screws. 


Y-LEYEL ADJUSTMENTS 


Eq. 60.] 


85 


If in error, the rod reading on the near point should be 
checked before making the final adjustment. 

6. Vernier of Vertical Circle. — With the bubble of the 
telescope level centered, and the plates carefully leveled, set 
the vernier of the vertical circle at zero. 

b. Y-Level Adjustments. 

1. Horizontal Wire. —The horizontal wire is made truly 
horizontal by a similar method to that used with the transit 
for the vertical wire (see above). 

2. Line of Collimation. — Sight on a well-defined point 
with the intersection of the wires after having loosened the 
clips so that the telescope is free to rotate in the Y’s. Both 
leveling screws and tangent screw may be used in making 
this pointing, as it is not essential that the telescope be level. 

Rotate the telescope 180° about its axis and correct one- 
half the apparent error, if any, by adjusting the reticule. 
Adjust one wire at a time until the intersection will remain on 
the point throughout the entire revolution. 

The point used for this adjustment will preferably be at 
about the distance* of the average length of sight. This ad¬ 
justment makes the line of collimation coincide with the axis 
of the rings. 

3. Level. — (a) Lateral. Level approximately and turn 
the telescope slightly in the Y’s. If the bubble remains fixed, 
the level is in the same plane with the axis of the rings. If 
not, adjust by the lateral screws at one end of the tube. 

(b) Vertical. Level up over both pairs of screws, then 
clamp over one of them. With the clips open, center the 
bubble accurately. Carefully remove the telescope from the 
Y’s and reverse and return it. Correct one-half the apparent 
error, if any, by the use of the adjusting screws. Relevel 
and repeat. 

This makes the level parallel with the bottom element of the 
rings and, if the rings are of equal size, with the line of colli¬ 
mation. This is therefore an indirect or mechanical way of 
making the line of collimation parallel with the level and, 
especially with an unknown instrument, it should be checked 
by the direct or peg method. 

4. Y’s. — For convenience the Y’s should be adjusted to 
nearly the same length, so that the level tube will be perpen- 


86 


RAILROAD SURVEYING 


[§ 56 . 

dicular to the axis of rotation. Level over both pairs of 
screws; finally over one and swing 180° in azimuth. Correct 
one-half the error, if any, by adjusting the Y’s. 

c. Dumpty Level Adjustments. 

1. Horizontal Wire. —The same as for the Y level. 

2. Level. —* The level is made perpendicular to the axis of 
rotation by the same method as was used for the Y’s of the 
Y-level, except that the level is adjusted, as the standards 
supporting the telescope are usually not adjustable. 

3. Line of Collimation. — Use the two-peg method, adjust¬ 
ing the wire instead of the bubble as in the transit. 

The above are the ordinary methods of adjustment which 
assume that the line of collimation is a fixed line. Since this 
line is determined by the wires and the optical center of the 
objective, its direction is not constant for different lengths of 
sight unless it is parallel with the object glass slide. 

To test this for the horizontal wire, as in leveling, the three- 
peg method is necessary. Select three points at about the 
same distance apart and determine the exact differences of 
elevation by taking readings for each pair from a point half¬ 
way between. Then setting up near one of the end points, 
take readings on all three. If the differences are the same as 
before, or if the discrepancies are proportional to the dis¬ 
tances from the instrument, the objective slide is parallel to 
the line of collimation; if not, the slide may be adjusted on 
some instruments, while on others the only remedy is to return 
the instrument to the maker. 

For testing the vertical wire of the transit, line in a series 
of pegs, making the first one 30 to 50 feet from the transit. 
These points should be in a straight line. Test by plunging 
and reversing, sighting on the first point, and seeing if the 
others are on line. 


PROBLEMS 


87 


PROBLEMS. 

1. On a certain line it is decided to elevate the outer rail on 
curves at the rate of £ of an inch per degree of curve. 

а. Find the chord whose middle ordinate will equal the required 
elevation. 

б. For what speed will this elevation be correct? 

2 . Find, by an exact method, the middle ordinates for curv¬ 
ing 32-foot rails to 20, 35 and 50 degree curves and compare your 
results with Table XII. 



3 . A track on 3-foot gage is to be elevated for a speed of 20 
miles per hour. 

а. Find the elevation in inches per degree of curve. 

б. Find the chord whose middle ordinate equals this elevation. 

4 . A+ 0.3 per cent gradient intersects a —0.7 per cent at 
sta. 424 + 50, with an elevation of 478.50. Find the grades at 
the regular stations for a vertical curve changing gradient at the 
rate of 0.1 per cent per station. 



88 


PROBLEMS 


5 . Given the data of problem 4. Find the grades at the regu¬ 
lar stations and + 50’s for a vertical curve changing gradient 
at the rate of 0.1 per cent per 50 feet. 

6. A 6° curve is elevated for a speed of 45 miles per hour. 
Find the length of transition curve required by the recommen¬ 
dations of the Am. Ry. Eng. and M. of W. Assoc. 

7 . How much is the track shortened by the appliqation of 
300-foot transitions to a 6° curve 6 stations long? 

8 . Find the tangent distances if a 300-foot transition be added 
at one end only of the curve of problem 6. 

9 . Compute coordinates for points 60 feet apart on a 360-foot 
transition to a 9° curve. 

10 . Apply formula (14) of Part II to the determination of the 
deflections for running in the transition of problem 8 from the 
PTC, from the PC 1 and for each half from its middle point. 

11 . In Fig. 23, A 1 is 28° and A 2 , 32°. The degree of the first 
curve is D x = 9° and of the second, D 2 = 5°. Find the tangent 
distances if a 180-foot transition is used at the beginning of the 
9° curve, a 160-foot between the two curves and a 250-foot at the 
end of the 5° curve. 

Suggestion. — Use the offsets and long chords as the courses 
of a traverse in computing the coordinates of I referred to the first 
tangent as an axis and D as an origin, i.e., compute DEFGHI as a 
traverse. 

12 . Compute the tangents for the compound curve of the pre¬ 
ceding problem if no transition curves are used. * 




































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































OCT 20 190$ 





























































































